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Versions: 00 01 03 04 05 rfc2676                                        
Internet Engineering Task Force               R. Guerin/S. Kamat/A. Orda
INTERNET DRAFT                                          IBM/IBM/Technion
                                               T. Przygienda/D. Williams
                                                              Lucent/IBM
                                                         30 January 1998


               QoS Routing Mechanisms and OSPF Extensions
                  draft-guerin-qos-routing-ospf-03.txt


Status of This Memo

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   documents of the Internet Engineering Task Force (IETF), its Areas,
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Abstract

   This memo describes extensions to the OSPF [Moy97] protocol to
   support QoS routes.  The focus of this document is on the algorithms
   used to compute QoS routes and on the necessary modifications to OSPF
   to support this function, e.g., the information needed, its format,
   how it is distributed, and how it is used by the QoS path selection
   process.  Aspects related to how QoS routes are established and
   managed are also briefly discussed.  The goal of this document is
   to identify a framework and possible approaches to allow deployment
   of QoS routing capabilities with the minimum possible impact to the
   existing routing infrastructure.











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                                Contents



Status of This Memo                                                    i

Abstract                                                               i

 1. Introduction                                                       1
     1.1. Overall Framework . . . . . . . . . . . . . . . . . . . .    1
     1.2. Simplifying Assumptions . . . . . . . . . . . . . . . . .    2

 2. Path Selection Information and Algorithms                          4
     2.1. Metrics . . . . . . . . . . . . . . . . . . . . . . . . .    4
     2.2. Advertisement of Link State Information . . . . . . . . .    5
     2.3. Path Selection Algorithms . . . . . . . . . . . . . . . .    6
           2.3.1. Exact Pre-Computed QoS Paths (BF) . . . . . . . .    7
           2.3.2. On-Demand Computation of QoS Paths (Dijkstra) . .   12
           2.3.3. Exact & Approximate Pre-Computed QoS paths
                          (Dijkstra) . . . . . . . . . . . . . . . .  13
     2.4. Extracting Forwarding Information from Routing Table  . .   16

 3. Establishment and Maintenance of QoS Routes                       16

 4. OSPF Protocol Enhancements                                        18
     4.1. QoS -- Optional Capabilities  . . . . . . . . . . . . . .   18
     4.2. Encoding Resources as Extended TOS  . . . . . . . . . . .   19
           4.2.1. Encoding bandwidth resource . . . . . . . . . . .   20
           4.2.2. Encoding Delay  . . . . . . . . . . . . . . . . .   23
     4.3. Packet Formats  . . . . . . . . . . . . . . . . . . . . .   23
     4.4. Calculating the Inter-area Routes . . . . . . . . . . . .   23
     4.5. Open Issues . . . . . . . . . . . . . . . . . . . . . . .   23

 A. Pseudocode for BF Algorithm                                       24

 B. Pseudocode for On-Demand Dijkstra Algorithm                       26

 C. Pseudocode for Precomputed Dijkstra Algorithm                     28

 D. Zero-Hop Edges                                                    30

 E. Explicit Routing Support                                          31

 F. Computational Complexity                                          33

 G. Extension:  Handling Propagation Delays                           35



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 H. Accounting for Link Metric Inaccuracy in Path Selection           36


















































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1. Introduction

   In this document we describe a set of proposed additions to the
   OSPF routing protocol (the additions are built on top of OSPF
   V2 [Moy97]) to support Quality-of-Service (QoS) routing in IP.
   In particular, we discuss the metrics required to support QoS,
   the associated link advertisement mechanisms, the path selection
   algorithm, as well as aspects of route establishment.  Our goals are
   to define an approach which while achieving the goals of improving
   performance for QoS flows (likelihood to be routed on a path capable
   of providing the requested QoS), does so with the least possible
   impact on the existing OSPF protocol.  Given the inherent complexity
   of QoS routing, achieving this goal obviously implies trading-off
   ``optimality'' for ``simplicity'', but we believe this to be required
   in order to facilitate deployment of QoS routing capabilities.


1.1. Overall Framework

   We consider a network (1) that supports both best-effort packets and
   packets with QoS guarantees.  The way in which the network resources
   are split between the two classes is irrelevant to our proposal,
   except for the assumption that each QoS capable router in the network
   is able to dedicate some of its resources to satisfy the requirements
   of QoS packets.  QoS capable routers are also assumed to be able to
   identify and advertise the amount of their resources that remain
   available for additional QoS flows.  In addition, we limit ourselves
   to the case where all the routers involved support the QoS extensions
   described in this document, i.e., we do not consider the problem
   of establishing a route in a heterogeneous environment where some
   routers are QoS-capable and others are not.  Furthermore, in this
   document we focus on the case of unicast flows, although many of the
   additions we define are applicable to multicast flows as well.

   We assume that a flow with QoS requirements will specify them
   in some fashion that is accessible to the routing protocol.  For
   example, this could correspond to the arrival of an RSVP [RZB+97]
   PATH message, whose TSpec is passed to routing together with the
   destination address.  After processing such a request, the routing
   protocol returns a path that it deems the most suitable given the
   flow's requirements.  Depending on the scope of the path selection
   process, this returned path could range from simply identifying the

----------------------------
1. In this document we commit the abuse of notation of calling a
   ``network'' the interconnection of routers and networks through which
   we attempt to compute a QoS path.




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   best next hop, i.e., traditional hop-by-hop routing, to specifying
   all intermediate nodes to the destination, i.e., an explicit route.
   Note that this decision impacts the operation of the path selection
   algorithm as it translates into different requirements in order
   to construct and return the appropriate path information.  Note
   also that extension to multicast paths will impact differently a
   hop-by-hop and an explicit route based approach.

   For simplicity, we will describe the path computation algorithm
   assuming hop-by-hop routing.  Extensions to support explicit routing
   are discussed in appendix E.

   In this document, we focus on the aspect of selecting an appropriate
   path based on information about link metrics and flow requirements.
   There are obviously many other aspects that need to be specified in
   order to define a complete proposal for QoS routing.  For example,
   we incorporate a rather simplistic high level admission control
   policy based on the path length.  High level admission control is
   important because often admitting a flow even when a feasible path
   is found is not desirable if it will result in an inefficient use of
   network resources.  Another aspect concerns controlling the overhead
   of additional link state updates caused by more frequent changes to
   link metrics without adversely affecting the performance of path
   selection.  We present a brief discussion of various alternatives
   that trade accuracy of link state information with protocol overhead
   in Section 2.2.  A discussion of some approaches to account for the
   metric inaccuracies in path selection is deferred to Appendix H.
   Management of QoS paths is yet another aspect of a complete solution.
   Specifically, once a suitable path has been identified for a flow, it
   may be desirable to keep the path assigned (pinned) to the flow as
   long as it is deemed adequate in order to avoid frequent oscillations
   and routing instability.  A brief discussion of path management is
   provided in Section 3 and the reader is referred to [GKH97] for
   further details of one solution to this problem.


1.2. Simplifying Assumptions

   In order to achieve our goal of a minimal impact to the existing
   protocol, we impose certain restrictions on the range of requirements
   the QoS path selection algorithm needs to deal with directly.
   Specifically, a policy scheme is used to a priori prune from
   the network, those portions that would be unsuitable given the
   requirements of the flow.  This limits the ``optimization'' performed
   by the path selection to a containable set of parameters, which helps
   keep complexity at an acceptable level.  Specifically, the path
   selection algorithm will focus on selecting a path that is capable of
   satisfying the bandwidth requirement of the flow, while at the same



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   time trying to minimize the amount of network resources that need to
   be allocated to support the flow, i.e., minimize the number of hops
   used.

   This focus on bandwidth is adequate in most instances, but does not
   fully capture the complete range of potential QoS requirements.  For
   example, a delay-sensitive flow of an interactive application could
   be put on a path using a satellite link, if that link provided a
   direct path and had plenty of unused bandwidth.  This would clearly
   be an undesirable choice.  Our approach to preventing such poor
   choices, is to assign delay-sensitive flows to a policy that would
   eliminate from the network all links with high propagation delay,
   e.g., satellite links, before invoking the path selection algorithm.
   In general, each existing policy would present to the path selection
   algorithm its correspondingly pruned network topology, and the same
   algorithm would then be used to generate an appropriate path.

   Another important aspect in minimizing the impact of QoS routing
   is to develop a solution that has the smallest possible computing
   overhead.  Additional computations are unavoidable, but it is
   desirable to keep the total cost of QoS routing at a level comparable
   to that of traditional routing algorithms.  In this document, we
   describe several alternatives to the path selection algorithm,
   that represent different trade-offs between simplicity, accuracy,
   and computational cost.  In particular, we specify algorithms
   that generate exact solutions based either on pre-computations or
   on-demand computations.  We also describe algorithms that allow
   pre-computations at the cost of some loss in accuracy, but with
   possibly lower complexity or greater ease of implementation.  It
   should be mentioned, that while several alternative algorithms are
   described in this document, the same algorithm needs to be used
   consistently within a given routing domain.  This requirement can
   be relaxed when explicit routing is used as the responsibility
   of selecting a QoS path lies with a single entity, the origin of
   the request, which ensures consistency.  Hence, it may then be
   possible for each router to use a different path selection algorithm.
   However, in general, the use of a common path selection algorithm is
   recommended, if not necessary, for proper operation.

   The rest of this document is structured as follows.  In Section 2,
   we describe the path computation process and the information it
   relies on.  In Section 3, we briefly review some issues associated
   with path management and their implications.  In Section 4, we go
   over the extensions to OSPF that are needed in order to support the
   path selection process of Section 2.  Finally, several appendices
   provide details on the different path selection algorithms described
   in Section 2 and outline several additional work items.




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2. Path Selection Information and Algorithms

   This section describes several path selection algorithms that
   can be used to generate QoS capable paths based on different
   trade-offs between accuracy, computational complexity, and ease of
   implementation.  In addition, the section also covers aspects related
   to the type of information, i.e., metrics, on which the algorithms
   operate, and how that information is made available, i.e., link state
   advertisements.  The discussion on these topics is of a generic
   nature, and OSPF specific details are provided in Section 4.


2.1. Metrics

   As stated earlier, the process of selecting a path that can satisfy
   the QoS requirements of a new flow relies on both the knowledge of
   the flow's requirements and characteristics, and information about
   the availability of resources in the network.  In addition, for
   purposes of efficiency, it is also important for the algorithm to
   account for the amount of resources the network has to allocate in
   order to support a new flow.  In general, the network prefers to
   select the ``cheapest'' path among all paths suitable for a new flow.
   Furthermore, the network may also decide not to accept a new flow
   for which it identified a feasible path, if it deems the cost of the
   path to be too high.  Accounting for these aspects involves several
   metrics on which the path selection process is based.  They include:

    -  Link available bandwidth:  As mentioned earlier, we assume that
       most QoS requirements are derivable from a rate-related quantity,
       termed ``bandwidth''.  We further assume that associated with
       each link is a maximal bandwidth value, e.g., the link physical
       bandwidth or some fraction thereof that has been set aside for
       QoS flows.  Since for a link to be capable of accepting a new
       flow with given bandwidth requirements, at least that much
       bandwidth must be still available on the link, the relevant link
       metric is, therefore, the (current) amount of available (i.e.,
       unallocated) bandwidth.

    -  Hop-count:  This quantity is used as a measure of the path cost
       to the network.  A path with a smaller number of hops (that can
       support a requested connection) is typically preferable, since
       it consumes fewer network resources.  While, as a general rule,
       each edge in the graph on which the path is computed should be
       counted as one hop, some edges, specifically those that connect
       a transit network to a router, should not be taken into account.
       (See Appendix D for a detailed explanation.)





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    -  Policy:  As previously discussed, policies are used to identify
       the set of links in the network that need to be considered when
       running the path selection algorithm.  In particular, policies
       are used to prune from the network links whose performance or
       other characteristics are incompatible with the requirements of a
       flow.

       A specific policy example of special importance is the
       elimination of high latency links when considering path selection
       for delay sensitive flows.  The use of policies to handle
       specific requirements allows considerable simplification in
       the optimization task to be performed by the path selection
       algorithm.


2.2. Advertisement of Link State Information

   It is assumed that each router maintains an updated database of the
   network topology, including the current state (available bandwidth)
   of each link.  As mentioned, the distribution of link state (metrics)
   information is based on extending OSPF mechanisms.  However, in
   addition to how link state information is distributed, another
   important aspect is when such distribution is to take place.

   One option is to mandate periodic updates, where the period of
   updates is determined based on a tolerable corresponding load on the
   network and the routers.  The main disadvantage of such an approach
   is that major changes in the bandwidth available on a link could
   remain unknown for a full period and, therefore, result in many
   incorrect routing decisions.  Ideally, one would want routers to have
   the most current view of the bandwidth available on all links in the
   network, so that they can make the most accurate decision on which
   path to select.  Unfortunately, this then calls for very frequent
   updates, e.g., close to every time the available bandwidth of a link
   changes, which is neither scalable nor practical.

   In general, we are faced with a trade-off between the protocol
   overhead of frequent updates and the accuracy of the network state
   information that the path selection algorithm depends on.  We
   outline below a few possible link state update policies that strike a
   practical compromise.

   The basic idea is to trigger link state advertisements only when
   there is a significant change in the value of metrics since the last
   advertisement.  The notion of significance of a change can be based
   on an ``absolute'' scale or a ``relative'' one.  An absolute scale
   means partitioning the range of values that a metric can take into
   equivalence classes and triggering an update whenever the metric



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   changes sufficiently to cross a class boundary (2).  A relative
   scale, on the other hand, triggers updates when the percentage change
   in the metric value exceeds a predefined threshold.

   In either of the above two approaches, the value of a metric
   advertised in an LSA could either be the actual value or a quantized
   one according to some rule.  Thus, these design decisions introduce
   a certain degree of inaccuracy where an advertised value of a link
   metric implicitly indicates a range of potential current values
   of the metric.  The path selection algorithm that we describe in
   the next subsection operates on the advertised available bandwidth
   values.  Potential enhancements to the path selection algorithm
   that seek to account for the inaccuracies in link metrics that are
   introduced due to the update trigger policies will be described in
   Appendix H.

   Even though the update triggering mechanisms described above seek to
   reduce protocol traffic by not advertising small changes to metrics,
   the only direct means of controlling this overhead is through a hold
   down timer that enforces a minimum spacing between two successive
   updates.  This introduces an additional degree of inaccuracy in the
   topology database where even the boundaries of the potential range
   of values for a given advertised metric value become fuzzy.  Further
   research is needed to effectively account for such inaccuracies
   during path selection.


2.3. Path Selection Algorithms

   There are several aspects to the path selection algorithms.  The
   main ones include the optimization criteria it relies on, the exact
   topology on which it is run, and when it is invoked.

   As mentioned before, invocation of the path selection algorithm can
   be either per flow setup, or when warranted by changes in link states
   when the algorithm used allows precomputation of paths (more on this
   below).

   The topology on which the algorithm is run is, as with the standard
   OSPF path selection, a directed graph where vertices (3) consist of
   routers and networks (transit vertices) as well as stub networks
   (non-transit vertices).  When computing a path, stub networks are

----------------------------
2. Some hysteresis mechanism can be added to suppress updates when the
   metric value oscillates around a class boundary.
3. In this document, we use the terms node and vertex interchangeably.




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   added as a post-processing step, which is essentially similar to
   what is done with the current OSPF routing protocol.  In addition,
   for each policy supported on a router, the topology used by the
   path selection algorithm is correspondingly pruned to reflect the
   constraints imposed by the policy, and in some instances the flow
   requirements.

   The optimization criteria used by the path selection are reflected
   in the costs associated with each interface in the topology and how
   those costs are accounted for in the algorithm itself.  As mentioned
   before, the cost of a path is a function of both its hop count and
   the amount of available bandwidth.  As a result, each interface
   has associated with it a metric, that corresponds to the amount of
   bandwidth which remains available on this interface.  This metric
   is combined with hop count information to provide a cost value,
   in a manner that depends on the exact form of the path selection
   algorithm.  It will, therefore, be detailed in the corresponding
   sections below, but all the different alternatives that are described
   share a common goal.  That is, they all aim at picking a path with
   the minimum possible number of hops among those that can support
   the requested bandwidth.  When several such paths are available,
   the preference is for the path whose available bandwidth (i.e., the
   smallest value on any of the links in the path) is maximal.  The
   rationale for the above rule is the following:  we focus on feasible
   paths (as accounted by the available bandwidth metric) that consume
   a minimal amount of network resources (as accounted by the hop-count
   metric); and the rule for selecting among these paths aims at
   balancing load as well as maximizing the likelihood that the required
   bandwidth will indeed be available.

   It should be noted that standard routing algorithms are typically
   single objective optimizations, i.e., they may minimize the
   hop-count, or maximize the path bandwidth, but not both.  Double
   objective path optimization is a more complex task, and, in
   general, it is an intractable problem [GJ79].  Nevertheless, as
   we will see, because of the specific nature of the two objectives
   being optimized (bandwidth and hop count), the complexity of our
   proposed algorithm(s) is competitive with even that of standard
   single-objective algorithms.  For readers interested in a thorough
   treatment of the topic, connected with insights into the connection
   between the different algorithms, linear algebra and modification of
   metrics, [Car79] is recommended.


2.3.1. Exact Pre-Computed QoS Paths (BF)

   In this section, we describe a path selection algorithm, that for a
   given network topology and link metrics (available bandwidth) allows



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   us to pre-compute all possible QoS paths, and also has a reasonably
   low computational complexity.  Specifically, the algorithm allows
   us to pre-compute for any destination a minimum hop count path with
   maximum bandwidth, and has a computational complexity comparable to
   that of a standard shortest path algorithm (4).

   The path selection algorithm is based on a Bellman-Ford (BF) shortest
   path algorithm that is adapted to compute paths of maximum available
   bandwidth for all hop counts.  It is a property of the BF algorithm
   that, at its h-th iteration, it identifies the optimal (in our
   context:  maximal bandwidth) path between the source and each
   destination, among paths of at most h hops.  In other words, the
   cost of a path is a function of its available bandwidth, i.e., the
   smallest available bandwidth on all links of the path, and finding
   a minimum cost path amounts to finding a maximum bandwidth path.
   However, we also take advantage of the fact that the BF algorithm
   progresses by increasing hop count, to essentially get for free the
   hop count of a path as a second optimization criteria.

   Specifically, at the kth (hop count) iteration of the algorithm,
   the maximum bandwidth available to all destinations on a path of
   no more than k hops is recorded (together with the corresponding
   routing information).  After the algorithm terminates, this
   information enables us to identify for all destinations and bandwidth
   requirements, the path with the smallest possible number of hops and
   sufficient bandwidth to accommodate the new request.  Furthermore,
   this path is also the one with the maximal available bandwidth among
   all the feasible paths with at most these many hops.  This is because
   for any hop count, the algorithm always selects the one with maximum
   available bandwidth.

   We now proceed with a more detailed description of the algorithm
   and the data structure used to record routing information, i.e.,
   the QoS routing table that gets built as the algorithm progresses
   (pseudo-code for the algorithm can be found in Appendix A).  As
   mentioned before, the algorithm operates on a directed graph
   consisting only of transit vertices (routers and networks), with
   stub-networks subsequently added to the path(s) generated by the
   algorithm.  The metric associated with each edge in the graph is the
   bandwidth available on the corresponding interface.  Let us denote
   by bn;mthe available bandwidth on the edge between vertices n and
   m.  The vertex corresponding to the router where the algorithm is
   being run, i.e., the computing router, is denoted as the ``source

----------------------------
4. See Appendix F for a more comprehensive discussion on the aspect of
   computational complexity.




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   node'' for the purpose of path selection.  The algorithm proceeds to
   pre-compute paths from this source node to all possible destination
   networks and for all possible bandwidth values.  At each (hop count)
   iteration, intermediate results are recorded in a QoS routing table,
   which has the following structure:

The QoS routing table:

    -  a Kx H matrix, where K is the number of destinations (vertices
       in the graph) and H is the maximal allowed (or possible) number
       of hops for a path.

    -  The (n;h) entry is built during the hth iteration (hop count
       value) of the algorithm, and consists of two fields:

        *  bw:  the maximum available bandwidth, on a path of at most h
           hops between the source node (router) and destination node
           n;

        *  neighbor:  this is the routing information associated with
           the h (or less) hops path to destination node n, whose
           available bandwidth is bw.  In the context of hop-by-hop
           path selection (5), the neighbor information is simply the
           identity of the node adjacent to the source node on that
           path.  As a rule, the ``neighbor'' node must be a router and
           not a network (see Appendix D), the only exception being
           the case where the network is the destination node (and the
           selected path is the single edge interconnecting the source
           to it).

   Next, we provide additional details on the operation of the algorithm
   and how the entries in the routing table are being updated as the
   algorithm proceeds.  For simplicity, we first describe the simpler
   case where all edges count as ``hops'', and later explain how
   zero-hop edges (see Appendix D for further details) are handled.

   When the algorithm is invoked, the routing table is first initialized
   with all bw fields set to 0 and neighbor fields cleared.  Next, the
   entries in the first column (which corresponds to one-hop paths) of
   the neighbors of the computing router are modified in the following
   way:  the bw field is set to the value of the available bandwidth
   on the direct edge from the source.  The neighbor field is set to

----------------------------
5. Modifications to support explicit routing are discussed in
   Appendix E.





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   the identity of the neighbor of the computing router, i.e., the next
   router on the selected path.

   Afterwards, the algorithm iterates for at most H iterations
   (considering the above initial iteration as the first).  H can be
   either the maximum possible hop count of any path, i.e., an implicit
   value, or it can be set explicitly in order to limit path lengths to
   some maximum value (6) to better control worst case complexity.

   At iteration h, we first copy column h-1  into column h.  In
   addition, the algorithm keeps a list of nodes that changed their
   bw value in the previous iteration, i.e., during the (h   -    1)-th
   iteration.  The algorithm then looks at each link (n;m) where n is
   a node whose bw value changed in the previous iteration, and checks
   the maximal available bandwidth on an (at most) h-hop path to node
   m whose final hop is that link.  This amounts to taking the minimum
   between the bw field in entry (n;h   -   1) and the link metric value
   bn;mkept in the topology database.  If this value is higher than the
   present value of the bw field in entry (m;h), then a better (larger
   bw value) path has been found for destination m and with at most h
   hops.  The bw field of entry (m;h) is then updated to reflect this
   new value.  In the case of hop-by-hop routing, the neighbor field
   of entry (m;h) is set to the same value as in entry (n;h  - 1).  This
   records the identity of the first hop (next hop from the source) on
   the best path identified thus far for destination m and with h (or
   less) hops.

   We conclude by outlining how zero-hop edges are handled.  At each
   iteration h (starting with the first), whenever an entry (m;h) is
   modified, it is checked whether there are zero-cost edges (m;k)
   emerging from node m, which is the case when m is a transit network
   (see Appendix D).  In that case, we attempt to improve the entry of
   node k that corresponds to the h-th hop, i.e., entry (k;h) (rather
   than entry (k;h  +   1)), since the edge (m;k) should not count as an
   additional hop.  As with the regular operation of the algorithm, this
   amounts to taking the minimum between the bw field in entry (m;h)
   and the link metric value bm;kkept in the topology database.  If
   this value is higher than the present value of the bw field in entry
   (k;h), then the bw field of entry (k;h) is updated to this new value.
   In the case of hop-by-hop routing, the neighbor field of entry (k;h)
   is set, as usual, to the same value as in entry (m;h) (which is also
   the value in entry (n;h- 1)).

----------------------------
6. This maximum value should be larger than the length of the minimum
   hop-count path to any node in the graph.





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   Note that while for simplicity of the exposition, the issue of equal
   cost, i.e., same hop count and available bandwidth, is not detailed
   in the above description, it is straightforward to add this support.
   It only requires that the neighbor field be expanded to record the
   list of next (previous) hops, when multiple equal cost paths are
   present.

Addition of Stub Networks

   As was mentioned earlier, the path selection algorithm is run
   on a graph whose vertices consist only of routers and transit
   networks and not stub networks.  This is intended to keep the
   computational complexity as low as possible as stub networks can
   be added relatively easily through a post-processing step.  This
   second processing step is similar to the one used in the current OSPF
   routing table calculation [Moy97], with some differences to account
   for the QoS nature of routes.

   Specifically, after the QoS routing table has been constructed, all
   the router vertices are again considered.  For each router, stub
   networks whose link appears in the router's link advertisements will
   be processed to determine QoS routes available to them.  The QoS
   routing information for a stub network is similar to that of routers
   and transit networks and consists of an extension to the QoS routing
   table in the form of an additional row.  The columns in that new row
   again correspond to paths of different hop counts, and contain both
   bandwidth and next hop information.  We also assume that an available
   bandwidth value has been advertised for the stub network.  As before,
   how this value is determined is beyond the scope of this document.
   The QoS routes for a stub network S are constructed as follows:

   Each entry in the row corresponding to stub network S has its bws
   field initialized to zero and its neighbor set to null.  When stub
   network S is found in the link advertisement of router V, the value
   bw(S,h) in the hth column of the row corresponding to stub network S
   is updated as follows:

   bw(S,h) = min ( bw(S,h) ; min ( bw(V,h) , b(V,S) ) ),

   where bw(V,h) is the bandwidth value of the corresponding column
   for the QoS routing table row associated with router V, i.e.,
   the bandwidth available on an h hop path to V, and b(V,S) is the
   advertised available bandwidth on the link from V to S.  The above
   expression essentially states that the bandwidth of a h hop paths to
   stub network S is updated using a path through router V, only if the
   minimum of the bandwidth of the h hop path to V and the bandwidth on
   the link between V and S is larger than the current value.




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   Update of the neighbor field proceeds similarly whenever the
   bandwidth of a path through V is found to be larger than or equal
   to the current value.  If it is larger, then the neighbor field
   of V in the corresponding column replaces the current neighbor
   field of S.  If it is equal, then the neighbor field of V in the
   corresponding column is concatenated with the existing field for S,
   i.e., the current set of neighbors for V is added to the current set
   of neighbors for S.


2.3.2. On-Demand Computation of QoS Paths (Dijkstra)

   In the previous section, we described an algorithm that allows
   pre-computation of QoS routes.  However, it may be feasible in
   some instances, e.g., limited number of requests for QoS routes,
   to instead perform such computations on-demand, i.e., upon receipt
   of a request for a QoS route.  The benefit of such an approach is
   that depending on how often recomputation of pre-computed routes is
   triggered, on-demand route computation can yield better routes by
   using the most recent link metrics available.  Another benefit of
   on-demand path computation is the associated storage saving, i.e.,
   there is no need for a QoS routing table.  This is essentially the
   standard trade-off between memory and processing cycles.

   In this section, we briefly describe how a standard Dijkstra
   algorithm can, for a given destination and bandwidth requirement,
   generate a minimum hop path that can accommodate the required
   bandwidth and also has maximum bandwidth.  Because the Dijkstra
   algorithm is already used in the current OSPF route computation,
   only differences from the standard algorithm are described.  Also,
   while for simplicity we do not consider here zero-hop edges (see
   Appendix D), the modification required for supporting them is
   straightforward.

   The algorithm essentially performs a minimum hop path computation,
   on a graph from which all edges, whose available bandwidth is less
   than that requested by the flow triggering the computation, have been
   removed.  This can be performed either through a pre-processing step,
   or while running the algorithm by checking the available bandwidth
   value for any edge that is being considered (pseudo-code for the
   algorithm can be found in Appendix B).  Another modification to a
   standard Dijkstra based minimum hop count path computation, is that
   the list of equal cost next (previous) hops which is maintained as
   the algorithm proceeds, needs to be sorted according to available
   bandwidth.  This is to allow selection of the minimum hop path with
   maximum available bandwidth.  Alternatively, the algorithm could also
   be modified to, at each step, only keep among equal hop count paths
   the one with maximum available bandwidth.  This would essentially



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   amount to considering a cost that is function of both hop count and
   available bandwidth.


2.3.3. Exact & Approximate Pre-Computed QoS paths (Dijkstra)

   This section outlines a Dijkstra-based algorithm that allows
   pre-computation of QoS routes for all destinations and bandwidth
   values.  The benefit of using a Dijkstra-based algorithm is a
   greater synergy with existing OSPF implementations.  paths is to
   consecutively compute shortest path spanning trees starting from
   a complete graph and removing links with less bandwidth than the
   threshold used in the previous computation.  This yields paths with
   possibly better bandwidth but of course more hops.  Despite large
   number of Dijkstra computations involved, several optimizations such
   as incremental spanning tree computation can be used and allow for
   efficient implementations in terms of complexity as well as storage
   since the structure of computed paths leans itself towards path
   compression [ST83].  Details including measurements and applicability
   studies can be found in [Prz95].

   A variation of this theme is to trade the ``accuracy'' of the
   pre-computed paths, (i.e., the paths being generated may be of a
   larger hop count than needed) for the benefit of using a modified
   version of Dijkstra shortest path algorithm and also saving on some
   computations.  This loss in accuracy comes from the need to rely on
   quantized bandwidth values, that are used when computing a minimum
   hop count path.  In other words, the range of possible bandwidth
   values that can be requested by a new flow is mapped into a fixed
   number of quantized values, and minimum hop count paths are generated
   for each quantized value.  For example, one could assume that
   bandwidth values are quantized as low, medium, and high, and minimum
   hop count paths are computed for each of these three values.  A new
   flow is then assigned to the minimum hop path that can carry the
   smallest quantized value, i.e., low, medium, or high, larger than or
   equal to what it requested.  We restrict our discussion here to this
   ``quantized'' version of the algorithm and present its pseudo-code in
   Appendix C.

   Here too, we discuss the elementary case where all edges count as
   ``hops'', and note that the modification required for supporting
   zero-hop edges is straightforward.

   As with the BF algorithm, the algorithm relies on a routing table
   that gets built as the algorithm progresses.  The structure of the
   routing table is as follows:

The QoS routing table:



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    -  a K x Q matrix, where K is the number of vertices and Q is the
       number of quantized bandwidth values.

    -  The (n;q) entry contains information that identifies the
       minimum hop count path to destination n, that is capable of
       accommodating a bandwidth request of at least bw[q] (the qth
       quantized bandwidth value).  It consists of two fields:

        *  hops:  the minimal number of hops on a path between the
           source node and destination n, that can accommodate a
           request of at least bw[q] units of bandwidth.

        *  neighbor:  this is the routing information associated with
           the minimum hop count path to destination node n, whose
           available bandwidth is at least bw[q].  With a hop-by-hop
           routing approach, the neighbor information is simply the
           identity of the node adjacent to the source node on that
           path.

   The algorithm operates again on a directed graph where vertices
   correspond to routers and transit networks.  The metric associated
   with each edge in the graph is as before the bandwidth available on
   the corresponding interface, where bn;mis the available bandwidth
   on the edge between vertices n and m.  The vertex corresponding to
   the router where the algorithm is being run is selected as the source
   node for the purpose of path selection, and the algorithm proceeds to
   compute paths to all other nodes (destinations).

   Starting with the highest quantization index, Q, the algorithm
   considers the indices consecutively, in decreasing order.  For each
   index q, the algorithm deletes from the original network topology
   all links (n;m) for which bn;m< bw[q], and then runs on the remaining
   topology a Dijkstra-based minimum hop count algorithm  (7) between
   the source node and all other nodes (vertices) in the graph.  Note
   that as with the Dijkstra used for on-demand path computation, the
   elimination of links such that bn;m  <  bw[q] could also be performed
   while running the algorithm.

   After the algorithm terminates, the q-th column in the routing table
   is updated.  This amounts to recording in the hops field the hop
   count value of the path that was generated by the algorithm, and by
   updating the neighbor field.  As before, the update of the neighbor

----------------------------
7. Note that a Breadth-First-Search (BFS) algorithm
   [CLR90] could also be used.  It has a lower complexity, but would not
   allow reuse of existing code in an OSPF implementation.




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   field depends on the scope of the path computation.  In the case
   of a hop-by-hop routing decision, the neighbor field is set to the
   identity of the node adjacent to the source node (next hop) on the
   path returned by the algorithm.  However, note that in order to
   ensure that the path with the maximal available bandwidth is always
   chosen among all minimum hop paths that can accommodate a given
   quantized bandwidth, a slightly different update mechanism of the
   neighbor field needs to be used in some instances.  Specifically,
   when for a given row, i.e., destination node n, the value of the
   hops field in column q is found equal to the value in column q  +  1
   (here we assume q  <  Q), i.e., paths that can accommodate bw[q] and
   bw[q+ 1] have the same hop count, then the algorithm copies the value
   of the neighbor field from entry (n;q+ 1) into that of entry (n;q).

Addition of Stub Networks

   This proceeds in a manner very similar to that of Section 2.3.1,
   except for some minor variations reflecting differences in the
   structure of the QoS routing table.  Specifically, the columns of
   the QoS routing table now correspond to quantized bandwidth values,
   and the bw field of a column entry has been replaced by a hops
   field.  Hence, the QoS routes for a stub network S are constructed
   as follows:

   Each entry in the row corresponding to stub network S has its hops
   field initialized to zero and its neighbor set to null.  When stub
   network S is found in the link advertisement of router V, the value
   hops(S,q) in the qth column of the row corresponding to stub network
   S is updated as follows:

   hops(S,q) = hops(V,q) IF (hops(V,q) <= hops(S,q) AND b(V ,S) >=
   bw[q]),

   where bw[q] is the qth quantized bandwidth value, and b(V,S) is
   the advertised available bandwidth on the link from V to S.  The
   above expression essentially states that the hop count of the path
   to stub network S capable of supporting a bandwidth allocation
   of bw[q], is updated using a path through router V, only if the
   corresponding path through V has fewer hops than the current one,
   and the bandwidth on the link between V and S is larger than bw[q].

   Update of the neighbor field proceeds similarly whenever the path
   through router V capable of supporting a bandwidth allocation of
   bw[q], is found to yield a hop count smaller than or equal to the
   current value.  If it is smaller, then the neighbor field of V in
   the corresponding column replaces the current neighbor field of S.
   If it is equal, then the neighbor field of V in the corresponding
   column is concatenated with the existing field for S, i.e., the



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   current set of neighbors for V is added to the current set of
   neighbors for S.


2.4. Extracting Forwarding Information from Routing Table

   When the QoS paths are precomputed, the forwarding information for
   a flow with given destination and bandwidth requirement needs to be
   extracted from the routing table.  The case of hop-by-hop routing is
   much simpler compared to explicit routing.  This is because, only the
   next hop needs to be returned instead of an explicit route.

   Specifically, assume a new request to destination, say, d, and with
   bandwidth requirements B.  The index of the destination vertex
   identifies the row in the QoS routing table that needs to be checked
   to generate a path.  How the row is searched to identify a suitable
   path depends on which algorithm was used to construct the QoS routing
   table.  If the Bellman-Ford algorithm of Section 2.3.1 is used, the
   search proceeds by increasing index (hop) count until an entry is
   found, say at hop count or column index of h, with a value of the
   bw field which is equal to or larger than B.  This entry points
   to the initial information identifying the selected path.  If the
   Dijkstra algorithm of Section 2.3.3 is used, the first quantized
   value bBsuch that Bb     B  is first identified, and the associated
   column then determines the first entry in the QoS routing table that
   identifies the selected path.  The next hop information is then
   directly retrieved from the neighbor information of the first entry
   pointed to in the QoS routing table.

   If the path computation algorithm stores multiple equal cost paths,
   then some degree of load balancing can be achieved at the time
   of path selection.  A next hop from the list of equivalent next
   hops can be chosen in a round robin manner, or randomly with equal
   probability or randomly with a probability that is weighted by the
   actual available bandwidth on the local interface.

   The case of explicit routing is discussed in Appendix E.


3. Establishment and Maintenance of QoS Routes

   In this section, we briefly review issues related to how QoS paths
   are established and maintained.  For both, there are functional and
   protocol aspects that need to be covered.  We primarily address the
   functional aspects here and point to other references that address
   the protocol aspects.





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   The goal of QoS routing is to select paths for flows with QoS
   requirements, in such a manner as to increase the likelihood that the
   network will indeed be capable of satisfying them.  The use of QoS
   routing algorithms such as the ones described in this document have a
   number of implications above and beyond what is required when using
   standard routing algorithms.

   First, a specific mechanism needs to be used to identify flows with
   QoS requirements, so that they can be assigned to the corresponding
   QoS routing algorithm.  The RSVP protocol [RZB+97] can be used
   for this purpose.  Specifically, RSVP PATH messages serve as the
   trigger to query QoS routing.  Second, because of variations in
   the availability of resources in the network, routes between the
   same source and destination and for the same QoS, may often differ
   depending on when the request is made.  However, it is important
   to ensure that such changes are not always reflected on existing
   paths.  This is to avoid potential oscillations between paths and
   limit changes to cases where the initial selection turns out to be
   inadequate.

   As a result, some state information needs to be associated with a
   QoS route to determines its current validity, i.e., should the QoS
   routing algorithm be queried to generate a new and potentially better
   route, or does the current one remain adequate.  We say that a path
   is ``pinned'' when its state specifies that QoS routing need not be
   queried anew, while a path is considered ``unpinned'' otherwise.
   The main issue is then to define how, when, and where route pinning
   and unpinning is to take place.  In our context, where the RSVP
   protocol is used as the vehicle to request QoS routes, we also want
   this process to be as synergetic as possible with the existing RSVP
   state management.  In particular, our goal is to support pinning and
   unpinning of routes in a manner consistent with RSVP soft states
   while requiring minimal changes to the RSVP processing rules.

   It should be noted that some changes are unavoidable, especially
   to the interface between RSVP and routing.  Specifically,
   QoS routing requires, in addition to the current source and
   destination addresses, at a minimum, knowledge of the flow's traffic
   characteristics (TSpec), and possibly also service types (as per the
   information in the Adspec), PHOP, IP TTL value, etc.

   A broad RSVP-routing interface that enables this is described in
   [GKR97].  Use of such an interface in the context of reserving
   resources along an explicit path with RSVP is discussed in [GLG+97].
   Details of path management and a means of avoiding loops in case of
   hop-by-hop path set up can be found in [GKH97].





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4. OSPF Protocol Enhancements

   As stated above, a goal of this work is to limit the additions to the
   existing OSPF V2 protocol, while still providing the required level
   of support for QoS based routing.  To this end, all of the existing
   OSPF mechanisms, data structures, advertisements, and data formats
   remain in place.  The purpose of this section of the document is to
   list the enhancements to the OSPF protocol to support QoS as outlined
   in the previous sections.


4.1. QoS -- Optional Capabilities

   The OSPF Options field is present in OSPF Hello packets, Database
   Description packets and all LSAs.  The Options field enables OSPF
   routers to support (or not support) optional capabilities, and to
   communicate their capability level to other OSPF routers.  Through
   this mechanism, routers of differing capabilities can be mixed with
   an OSPF routing domain.  Currently, RFC 2178 [Moy97] specifies the
   following 5 bits in the options octet:


           +-----------------------------------------------+
           |  *  |  *  | DC  |  EA | N/P |  MC |  E  |  *  |
           +-----------------------------------------------+


   Note that the least significant bit (`T' bit) that was used to
   indicate TOS routing capability in the older OSPF specification
   [Moy94] has been removed.  In this context, the current OSPF
   specification [Moy97] states that:

   ``The TOS routing option has been deleted from OSPF. This action
   was required by the Internet standards process, due to lack of
   implementation experience with OSPF's TOS routing.  However, for
   backward compatibility the formats of OSPF's various LSAs remain
   unchanged, maintaining the ability to specify TOS metrics in
   router-LSAs, summary-LSAs, ASBR-summary-LSAs, and AS-external-LSAs.''

   We propose to reclaim the `T' bit as an indicator of router's QoS
   routing capability.  In fact, QoS capability can be viewed as an
   extension of the TOS-capabilities and QoS routing as a form of
   TOS-based routing.  A router sets this bit in its hello packets to
   indicate that it is capable of supporting such routing.  When this
   bit is set in a router or summary links link state advertisement, it
   means that there are QoS fields to process in the packet.  When this
   bit is set in a network link state advertisement it means that the
   network described in the advertisement is QoS capable.



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   We need to be careful in this approach so as to avoid confusing any
   old style (i.e., RFC 1583 based) TOS routing implementations.  The
   TOS metric encoding rules of QoS fields introduced further in this
   section will show how this is achieved.  Additionally, unlike the
   RFC 1583 specification that unadvertised TOS metrics be treated to
   have same cost as TOS 0, for the purpose of computing QOS routes,
   unadvertised TOS metrics (on a hop) indicate lack of connectivity for
   the specific TOS metrics (for that hop).


4.2. Encoding Resources as Extended TOS

   Introduction of QoS should ideally not influence the compatibility
   with existing OSPFv2 routers.  To achieve this goal, necessary
   extensions in packet formats must be defined in a way that either
   is understood by OSPFv2 routers, ignored or in the worst case
   ``gracefully'' misinterpreted.  Encoding of QoS metrics in the
   TOS field which fortunately enough is longer in OSPF packets
   than officially defined in [Alm92], allows us to mimic the new
   facility as extended TOS capability.  OSPFv2 routers will either
   disregard these definitions or consider those unspecified.  Specific
   precautions are taken to prevent careless OSPF implementations
   from influencing traditional TOS routing when misinterpreting the
   extension introduced.

   For QoS resources, 32 combinations are available through the use
   of the fifth bit in TOS fields contained in different LSAs.  Since
   [Alm92] defines TOS as being four bits long, this definition never
   conflicts with existing values.  Additionally, to prevent naive
   implementations that do not take all bits of the TOS field in OSPF
   packets into considerations, the definitions of the `QoS encodings'
   is aligned in their semantics with the TOS encoding.  Only bandwidth
   and delay are specified as of today and their values map onto
   `maximize throughput' and `minimize delay' if the most significant
   bit is not taken into account.  Accordingly, link reliability and
   jitter could be defined later if necessary.


        OSPF encoding   RFC 1349 TOS values
        ___________________________________________
        0               0000 normal service
        2               0001 minimize monetary cost
        4               0010 maximize reliability
        6               0011
        8               0100 maximize throughput
        10              0101
        12              0110
        14              0111



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        16              1000 minimize delay
        18              1001
        20              1010
        22              1011
        24              1100
        26              1101
        28              1110
        30              1111

        OSPF encoding   `QoS encoding values'
        -------------------------------------------

        32             10000
        34             10001
        36             10010
        38             10011
        40             10100 bandwidth
        42             10101
        44             10110
        46             10111
        48             11000 delay
        50             11001
        52             11010
        54             11011
        56             11100
        58             11101
        60             11110
        62             11111


        Representing TOS and QoS in OSPF.



4.2.1. Encoding bandwidth resource

   Given the fact that the actual metric field in OSPF packets only
   provides 16 bits to encode the value used and that links supporting
   bandwidth ranging into Gbits/s are becoming reality, linear
   representation of the available resource metric is not feasible.  The
   solution is exponential encoding using appropriately chosen implicit
   base value and number bits for encoding mantissa and the exponent.
   Detailed considerations leading to the solution described are not
   presented here but can be found in [Prz95].

   Given a base of 8, the 3 most significant bits should be reserved for
   the exponent part and the remaining 13 for the mantissa.  This allows




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   a simple comparison for two numbers encoded in this form, which is
   often useful during implementation.

   The following table shows bandwidth ranges covered when using
   different exponents and the granularity of possible reservations.


        exponent
        value x         range (2^13-1)*8^x      step 8^x
        -------------------------------------------------
        0               8,191                   1
        1               65,528                  8
        2               524,224                 64
        3               4,193,792               512
        4               33,550,336              4,096
        5               268,402,688             32,768
        6               2,147,221,504           262,144
        7               17,177,772,032          2,097,152

          Ranges of Exponent Values for 13 bits,
               base 8 Encoding, in Bytes/s



   The bandwidth encoding rule may be summarized as:  ``represent
   available bandwidth in 16 bit field as a 3 bit exponent (with assumed
   base of 8) followed by a 13 bit mantissa as shown below


        0       8       16
        |       |       |
        -----------------
       |EXP| MANT        |
        -----------------



   and advertise 2's complement of the above representation.''

   Thus, the above encoding advertises a numeric value that is

       2^16 -1 -(exponential encoding of the available bandwidth):

   This has the property of advertising a higher numeric value for lower
   available bandwidth, a notion that is consistent with that of cost.

   Although it may seem slightly pedantic to insist on the property
   that less bandwidth is expressed higher values, it has, besides



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   consistency, a robustness aspect in it.  A router with a poor OSPF
   implementation could misuse or misunderstand bandwidth metric as
   normal administrative cost provided to it and compute spanning trees
   with a ``normal'' Dijkstra.  The effect of a heavily congested link
   advertising numerically very low cost could be disastrous in such
   a scenario.  It would raise the link's attractiveness for future
   traffic instead of lowering it.  Evidence that such considerations
   are not speculative, but similar scenarios have been encountered, can
   be found in [Tan89].

   Concluding with an example, assume a link with bandwidth of 8
   Gbits/s = 1024^3 Bytes/s, its encoding would consist of an exponent
   value of 6 since 1024^3   =    4; 096 *  8^6, which would then have a
   granularity of 8^6    approx.    260 kBytes/s.  The associated binary
   representation would then be %(110) 0 1000 0000 0000% or 53,248 (8).
   The bandwidth cost (advertised value) of this link when it is idle,
   is then the 2's complement of the above binary representation,
   i.e., %(001) 1 0111 1111 1111% which corresponds to a decimal
   value of (2^16 -  1) -  53;248   =    12;287.  Assuming now a current
   reservation level of of 6;400 Mbits/s = 200  *   1024^2, there remains
   1;600 Mbits/s of available bandwidth on the link.  The encoding
   of this available bandwidth of 1'600 Mbits/s is 6;400   *  8^5, which
   corresponds to a granularity of 8^5  approx.   30 kBytes/s, and has a
   binary representation of %(101) 1 1001 0000 0000% or decimal value
   of 47,360.  The advertised cost of the link with this load level, is
   then %(010) 0 0110 1111 1111%, or (2^16-1) -47;360 = 18;175.

   Note that the cost function behaves as it should, i.e., the less
   bandwidth is available on a link, the higher the cost and the less
   attractive the link becomes.  Furthermore, the targeted property of
   better granularity for links with less bandwidth available is also
   achieved.  It should, however, be pointed out that the numbers given
   in the above examples match exactly the resolution of the proposed
   encoding, which is of course not always the case in practice.  This
   leaves open the question of how to encode available bandwidth
   values when they do not exactly match the encoding.  The standard
   practice is to round it to the closest number.  Because we are
   ultimately interested in the cost value for which it may be better
   to be pessimistic than optimistic, we choose to round costs up and,
   therefore, bandwidth down.

----------------------------
8. exponent in parenthesis








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4.2.2. Encoding Delay

   Delay is encoded in microseconds using the same exponential method
   as described for bandwidth except that the base is defined to be 4
   instead of 8.  Therefore the maximum delay that can be expressed is
   (2^13-1) *4^7 approx.134 seconds.


4.3. Packet Formats

   Given the extended TOS notation to account for QoS metrics, no
   changes in packet formats are necessary except for the introduction
   of Q-bit in the options field.  Routers not understanding the Q-bit
   should either not consider the QoS metrics distributed or consider
   those as `unknown' TOS.


4.4. Calculating the Inter-area Routes

   This document proposes a very limited use of OSPF areas, that is, it
   is assumed that summary links advertisements exist for all networks
   in the area.  This document does not discuss the problem of providing
   support for area address ranges and QoS metric aggregation.  This is
   left for further studies.


4.5. Open Issues

   Support for AS External Links, Virtual Links, and incremental updates
   for summary link advertisements are not addressed in this document
   and are left for further study.  For Virtual Links that do exist, it
   is assumed for path selection that these links are non-QoS capable
   even if the router advertises QoS capability.  Also, as stated
   earlier, this document does not address the issue of non-QoS routers
   within a QoS domain.



Acknowledgments

   We would like to thank the many people who have helped shape various
   aspects of this document and the approaches it describes, either
   through discussions or explicit suggestions.  In particular, we would
   like to acknowledge the help and inputs of John Moy, Dilip Kandlur,
   George Apostolopoulos and Dimitrios Pendarakis.






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                               APPENDICES


A. Pseudocode for BF Algorithm

Note:  The pseudocode below assumes a hop-by-hop forwarding approach in
   updating the neighbor field.  The modifications needed to support
   explicit route construction are straightforward.  The pseudocode
   also does not handle equal cost multi-paths for simplicity, but the
   modification needed to add this support are straightforward.

Input:
  V = set of vertices, labeled by integers 1 to N.
  L = set of edges, labeled by ordered pairs (n,m) of vertex labels.
  s = source vertex (at which the algorithm is executed).
  For all edges (n,m) in L:
    * b(n,m) = available bandwidth (according to last received update)
    on interface associated with the edge between vertices n and m.
    * If(n,m) outgoing interface corresponding to edge (n,m) when n is
      a router.
  H = maximum hop-count (at most the graph diameter).

Type:
  tab_entry: record
                 bw = integer,
                 neighbor = integer 1..N.

Variables:
  TT[1..N, 1..H]: topology table, whose (n,h) entry is a tab_entry record, such
                  that:
                    TT[n,h].bw is the maximum available bandwidth (as known
                      thus far) on a path of at most h hops between
                      vertices s and n,
                    TT[n,h].neighbor is the first hop on that path (a neighbor
                      of s). It is either a router or the destination n.

  S_prev: list of vertices that changed a bw value in the TT table
          in the previous iteration.
  S_new: list of vertices that changed a bw value (in the TT table
          etc.) in the current iteration.

The Algorithm:

begin;

  for n:=1 to N do  /* initialization */
  begin;
    TT[n,0].bw := 0;



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    TT[n,0].neighbor := null
    TT[n,1].bw := 0;
    TT[n,1].neighbor := null
  end;
  TT[s,0].bw := infinity;

  reset S_prev;
  for all neighbors n of s do
  begin;
    TT[n,1].bw := max( TT[n,1].bw, b[s,n]);
    if (TT[n,1].bw = b[s,n]) then TT[n,1].neighbor := If(s,n);
             /* need to make sure we are picking the maximum */
             /* bandwidth path for routers that can be reached */
             /* through both networks and point-to-point links */
       if (n is a router) then
           S_prev :=  S_prev union {n}
             /* only a router is added to S_prev, */
             /* if it is not already included in S_prev */
       else     /* n is a network: */
             /* proceed with network--router edges, without */
             /* counting another hop */
          for all (n,k) in L, k <> s, do
             /* i.e., for all other neighboring routers of n */
          begin;
          TT[k,1].bw := max( min( TT[n,1].bw, b[n,k]), TT[k,1].bw );
             /* In case k could be reached through another path */
             /* (a point-to-point link or another network) with */
             /* more bandwidth, we do not want to update TT[k,1].bw */
          if (min( TT[n,1].bw, b[n,k]) = TT[k,1].bw )
             /* If we have updated TT[k,1].bw by going through */
             /* network n  */
          then TT[k,1].neighbor := If(s,n);
             /* neighbor is interface to network n */
          if ( {k} not in S_prev) then S_prev :=  S_prev union {k}
             /* only routers are added to S_prev, but we again need */
             /* to check they are not already included in S_prev */
          end
  end;


  for h:=2 to H do   /* consider all possible number of hops */
  begin;
    reset S_new;
    for all vertices m in V do
    begin;
      TT[m,h].bw := TT[m,h-1].bw;
      TT[m,h].neighbor := TT[m,h-1].neighbor
    end;



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    for all vertices n in S_prev do
             /* as shall become evident, S_prev contains only routers */
    begin;
      for all edges (n,m) in L do
      if min( TT[n,h-1].bw, b[n,m]) > TT[m,h].bw then
      begin;
        TT[m,h].bw := min( TT[n,h-1].bw, b[n,m]);
        TT[m,h].neighbor := TT[n,h-1].neighbor;
        if m is a router then S_new :=  S_new union {m}
             /* only routers are added to S_new */
        else /* m is a network: */
             /* proceed with network--router edges, without counting them as */
             /* another hop */
        for all (m,k) in L, k <> n,
             /* i.e., for all other neighboring routers of m */
        if min( TT[m,h].bw, b[m,k]) > TT[k,h].bw then
        begin;
             /* Note: still counting it as the h-th hop, as (m,k) is a */
             /* network--router edge */
          TT[k,h].bw := min( TT[m,h].bw, b[m,k]);
          TT[k,h].neighbor := TT[m,h].neighbor;
          S_new :=  S_new union {k}
             /* only routers are added to S_new */
        end
      end
    end;
    S_prev := S_new;
            /* the two lists can be handled by a toggle bit */
    if S_prev=null then h=H+1   /* if no changes then exit */
  end;

end.



B. Pseudocode for On-Demand Dijkstra Algorithm

Note:  The pseudocode below assumes a hop-by-hop forwarding approach in
   updating the neighbor field.  The modifications needed to support
   explicit route construction are straightforward.  The pseudocode
   also does not handle equal cost multi-paths for simplicity, but the
   modifications needed to add this support have been described in
   section 2.3.2 and are straightforward.

Input:
  V = set of vertices, labeled by integers 1 to N.
  L = set of edges, labeled by ordered pairs (n,m) of vertex labels.
  s = source vertex (at which the algorithm is executed).



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  For all edges (n,m) in L:
    * b(n,m) = available bandwidth (according to last received update)
    on interface associated with the edge between vertices n and m.
    * If(n,m) = outgoing interface corresponding to edge (n,m) when n is
      a router.
  d = destination vertex.
  B = requested bandwidth for the flow served.

Type:
  tab_entry: record
                 hops = integer,
                 neighbor = integer 1..N,
                 ontree = boolean.

Variables:
  TT[1..N]: topology table, whose (n) entry is a tab_entry
                  record, such that:
                    TT[n].bw is the available bandwidth (as known
                        thus far) on a shortest-path between
                        vertices s and n,
                    TT[n].neighbor is the first hop on that path (a neighbor
                        of s). It is either a router or the destination n.
  S: list of candidate vertices;
  v: vertex under consideration;


The Algorithm:

begin;
  for n:=1 to N do  /* initialization */
  begin;
    TT[n].hops := infinity;
    TT[n].neighbor := null;
    TT[n].ontree := FALSE;
  end;
  TT[s].hops := 0;

  reset S;
  v:= s;
  while v <> d do
  begin;
    TT[v].ontree := TRUE;
    for all edges (v,m) in L and b(v,m) >= B do
    begin;
      if m is a router
      begin;
        if not TT[m].ontree then
        begin;



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          /* bandwidth must be fulfilled since all links >= B */
          if TT[m].hops > TT[v].hops + 1 then
          begin
            S := S union { m };
            TT[m].hops := TT[v].hops + 1;
            TT[m].neighbor := v;
          end;
        end;
      end;
      else /* must be a network, iterate over all attached routers */
      begin; /* each network -- router edge treated as zero hop edge */
        for all (m,k) in L, k <> v,
             /* i.e., for all other neighboring routers of m */
        if not TT[k].ontree and b(m,k) >= B then
        begin;
          if TT[k].hops > TT[v].hops  then
          begin;
            S := S union { k };
            TT[k].hops := TT[v].hops;
            TT[k].neighbor := v;
          end;
        end;
      end;
    end; /* of all edges from the vertex under consideration */

    if S is empty then
    begin;
      v=d; /* which will end the algorithm */
    end;
    else
    begin;
      v := first element of S;
      S := S - {v}; /* remove and store the candidate to consider */
    end;
  end; /* from processing of the candidate list */
end.



C. Pseudocode for Precomputed Dijkstra Algorithm

Note:  The pseudocode below assumes a hop-by-hop forwarding approach in
   updating the neighbor field.  The modifications needed to support
   explicit route construction are straightforward.  The pseudocode
   also does not handle equal cost multi-paths for simplicity, but
   the modification needed to add this support have been described in
   section 2.3.2 and are straightforward.




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Input:
  V = set of vertices, labeled by integers 1 to N.
  L = set of edges, labeled by ordered pairs (n,m) of vertex labels.
  s = source vertex (at which the algorithm is executed).
  bw[1..Q] = array of bandwidth values to ``quantize'' flow requests to.
  For all edges (n,m) in L:
    * b(n,m) = available bandwidth (according to last received update)
    on interface associated with the edge between vertices n and m.
    * If(n,m) = outgoing interface corresponding to edge (n,m) when n is
      a router.

Type:
  tab_entry: record
                 hops = integer,
                 neighbor = integer 1..N,
                 ontree = boolean.

Variables:
  TT[1..N, 1..Q]: topology table, whose (n,q) entry is a tab_entry
                  record, such that:
                    TT[n,q].bw is the available bandwidth (as known
                        thus far) on a shortest-path between
                        vertices s and n accommodating bandwidth b(q),
                    TT[n,q].neighbor is the first hop on that path (a neighbor
                        of s). It is either a router or the destination n.
  S: list of candidate vertices;
  v: vertex under consideration;
  q: ``quantize'' step

The Algorithm:

begin;
  for r:=1 to Q do
  begin;
    for n:=1 to N do  /* initialization */
    begin;
      TT[n,r].hops     := infinity;
      TT[n,r].neighbor := null;
      TT[n,r].ontree   := FALSE;
    end;
    TT[s,r].hops := 0;
  end;

  for r:=1 to Q do
  begin;
    S = {s};
    while S not empty do
    begin;



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      v := first element of S;
      S := S - {v}; /* remove and store the candidate to consider */
      TT[v,r].ontree := TRUE;
      for all edges (v,m) in L and b(v,m) >= bw[r] do
      begin;
        if m is a router
        begin;
          if not TT[m,r].ontree then
          begin;
            /* bandwidth must be fulfilled since all links >= bw[r] */
            if TT[m,r].hops > TT[v,r].hops + 1 then
            begin
              S := S union { m };
              TT[m,r].hops := TT[v,r].hops + 1;
              TT[m,r].neighbor := v;
            end;
          end;
        end;
        else /* must be a network, iterate over all attached
                routers */
        begin;
          for all (m,k) in L, k <> v,
               /* i.e., for all other neighboring routers of m */
          if not TT[k,r].ontree and b(m,k) >= bw[r] then
          begin;
            if TT[k,r].hops > TT[v,r].hops + 2 then
            begin;
              S := S union { k };
              TT[k,r].hops := TT[v,r].hops + 2;
              TT[k,r].neighbor := v;
            end;
          end;
        end;
      end; /* of all edges from the vertex under consideration */
    end; /* from processing of the candidate list */
  end; /* of ``quantize'' steps */
end.



D. Zero-Hop Edges

   The need to handle zero-hop edges is due to the potential presence
   of multiple access networks, e.g., T/R, E/N, or ATM, to interconnect
   routers.  Such entities are also represented by means of a vertex
   in the current OSPF operation.  Clearly, in such cases a network
   connecting two routers should be considered as a single hop path
   rather than a two hop path.  For example, consider three routers



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   A, B, and C connected over an Ethernet network N, which the OSPF
   topology represents as:

   In the above example, although there are directed edges in both
   directions, an edge from the network to any of the three routers
   must have zero ``cost'', so that it is not counted twice.  It should
   be noted that when considering such environments in the context
   of QoS routing, it is assumed that some entity is responsible
   for determining the ``available bandwidth'' on the network.  The
   specification of the operation of such an entity is beyond the scope
   of this document.


E. Explicit Routing Support

   As mentioned before, the scope of the path selection process can
   range from simply returning the next hop on the QoS path selected for
   the flow, to specifying the complete path that was computed, i.e.,
   an explicit route.  Obviously, the information being returned by the
   path selection algorithm differs in these two cases, and constructing
   it imposes different requirements on the path computation algorithm
   and the data structures it relies on.  While the presentation of
   the path computation algorithms focused on the hop-by-hop routing
   approach, the same algorithms can be applied to generate explicit
   routes with minor modifications.  These modifications and how they
   facilitate constructing explicit routes are discussed next.

   The general approach to facilitate construction of explicit routes
   is to update the neighbor field differently from the way it is done
   for hop-by-hop routing as described in Section 2.  Recall that in the
   path computation algorithms the neighbor field is updated to reflect
   the identity of the node adjacent to the source node on the partial
   path computed.  This facilitates returning the next hop at the
   source for the specific path.  In the context of explicit routing,
   the neighbor information is updated to reflect the identity of the
   previous router on the path.

   In general, there can be multiple equivalent paths for a given hop
   count.  Thus, the neighbor information is stored as a list rather
   than single value.  Associated with each neighbor, additional



 A----N----B
      |
      |
      C




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   information is stored to facilitate load balancing among these
   multiple paths at the time of path selection.  Specifically, we store
   the advertised available bandwidth on the link from the neighbor to
   the destination in the entry.

   With this change, the basic approach used to extract the complete
   list of vertices on a path from the neighbor information in the QoS
   routing table is to proceed `recursively' from the destination to
   the origin vertex.  The path is extracted by stepping through the
   precomputed QoS routing table from vertex to vertex, and identifying
   at each step the corresponding neighbor (precursor) information.  The
   process is described as recursive since the neighbor node identified
   in one step becomes the destination node for table look up in the
   next step.  Once the source router is reached, the concatenation of
   all the neighbor fields that have been extracted forms the desired
   explicit route.  This applies to algorithms of Sections 2.3.1 and
   2.3.3.  If at a particular stage there are multiple neighbor choices
   (due to equal cost multi-paths), one of them can be chosen at random
   with a probability that is weighted by the associated bandwidth on
   the link from the neighbor to the (current) destination.

   Specifically, assume a new request to destination, say, d, and with
   bandwidth requirements B.  The index of the destination vertex
   identifies the row in the QoS routing table that needs to be checked
   to generate a path.  How the row is searched to identify a suitable
   path depends on which algorithm was used to construct the QoS routing
   table.  If the Bellman-Ford algorithm of Section 2.3.1 is used, the
   search proceeds by increasing index (hop) count until an entry is
   found, say at hop count or column index of h, with a value of the bw
   field that is equal to or greater than B.  This entry points to the
   initial information identifying the selected path.  If the Dijkstra
   algorithm of Section 2.3.3 is used, the first quantized value bB
   such that bB B  is first identified, and the associated column then
   determines the first entry in the QoS routing table that identifies
   the selected path.

   Once this first entry has been identified, reconstruction of the
   complete list of vertices on the path proceeds similarly, whether
   the table was built using the algorithm of Sections 2.3.1 or 2.3.3.
   Specifically, in both cases, the neighbor field in each entry points
   to the previous node on the path from the source node and with the
   same bandwidth capabilities as those associated with the current
   entry.  The complete path is, therefore, reconstructed by following
   the pointers provided by the neighbor field of successive entries.

   In the case of the Bellman-Ford algorithm of Section 2.3.1, this
   means moving backwards in the table from column to column, using at
   each step the row index pointed to by the neighbor field of the entry



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   in the previous column.  Each time, the corresponding vertex index
   specified in the neighbor field is pre-pended to the list of vertices
   constructed so far.  Since we start at column h, the process ends
   when first column is reached, i.e., after h steps, at which point
   the list of vertices making up the path has been reconstructed.

   In the case of the Dijkstra algorithm of Section 2.3.3, the
   backtracking process is similar although slightly different because
   of the different relation between paths and columns in the routing
   table, i.e., a column now corresponds to a quantized bandwidth value
   instead of a hop count.  The backtracking now proceeds along the
   column corresponding to the quantized bandwidth value needed to
   satisfy the bandwidth requirements of the flow.  At each step, the
   vertex index specified in the neighbor field is pre-pended to the
   list of vertices constructed so far, and is used to identify the next
   row index to move to.  The process ends when an entry is reached
   whose neighbor field specifies the origin vertex of the flow.  Note
   that since there are as many rows in the table as there are vertices
   in the graph, i.e., N, it could take up to N steps before the
   process terminates.

   Note that the identification of the first entry in the routing table
   is identical to what was described for the hop-by-hop routing case.
   However, as described in this section, the update of the neighbor
   fields while constructing the QoS routing tables, is being performed
   differently in the explicit and hop-by-hop routing cases.  Clearly,
   two different neighbor fields can be kept in each entry and updates
   to both could certainly be performed jointly, if support for both
   explicit routing and hop-by-hop routing is needed.


F. Computational Complexity

   One generic aspect of the algorithmic complexity of computing
   QoS paths is the efficiency of the shortest path algorithm used.
   Specifically, in this document, we have described approaches based on
   both Bellman-Ford and Dijkstra shortest paths algorithms.  Dijkstra's
   algorithm has traditionally been considered more efficient for
   standard shortest path computations because of its lower worst case
   complexity.  However, the answer is not as simple as may appear, and
   in this section we briefly review a number of considerations, in
   particular in the context of multi-criteria QoS paths, which indicate
   that a BF approach may often provide a lower complexity solution.

   The asymptotic worst-case complexity of the Dijkstra algorithm is
   O(NlogN   +   M), where N is the number of vertices in the graph,
   and M the number of edges.  However, this bound is obtained
   under the assumption of a Fibonnaci heap implementation of the



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   Dijkstra algorithm, which is impractical due to the large constants
   involved [CLR90].  In practice, the Dijkstra algorithm is typically
   implemented using binary heaps, for which the asymptotic worst-case
   bound is O(MlogN).

   The asymptotic worst-case bound for the BF algorithm is O(H  .  M),
   where M is again the number of edges in the graph, and H, which is
   the maximum number of iterations of the algorithm, is an upper-bound
   on the number of hops in a shortest path.  Although, theoretically,
   H can be as large as N -  1, in practice it is usually considerably
   smaller than N.  Moreover, in some network scenarios an upper-bound
   U of small size (i.e., U   <<  N) is imposed on the allowed number
   of hops; for example, it might be decided to exclude paths that
   have more than, say, 16 hops, as part of a call admission scheme.
   In such cases, the number of iterations of the BF algorithm can be
   limited to U, thus bounding the number of operations to O(U  .  M),
   i.e., effectively to O(M).  As a consequence, as noted in [BG92],
   in practical networking scenarios, the BF algorithm can offer an
   efficient solution to the shortest path problem, one that often
   outperforms the Dijkstra algorithm. (9)

   In the context of QoS path selection, the potential benefits of the
   BF algorithm are even stronger.  As mentioned before, efficient
   selection of a suitable path for flows with QoS requirements cannot
   usually be handled using a single-objective optimization criterion.
   While multi-objective path selection is known to be an intractable
   problem [GJ79], the BF algorithm allows us to handle a second
   objective, namely the hop-count, which is reflective of network
   resources, at no additional cost in terms of complexity.  The
   Dijkstra algorithm requires some modifications (or approximations,
   e.g., bandwidth quantization) in order to be able to deal with hop
   count as a second objective.

   Therefore, in the context of a QoS path selection algorithm,
   where one objective is some QoS-oriented metric, such as available
   bandwidth, whereas the second is a hop-count metric, a BF-based
   algorithm provides an efficient scheme for pre-computing paths,
   i.e., one with a worst case asymptotic complexity of O(H    .   M).
   Alternatively, if QoS paths are pre-computed using a Dijkstra

----------------------------
9. For example, in the experimental comparison reported in [CGR94], the
   BF algorithm outperformed the Dijkstra algorithm in about one third
   of the studied types of topology, and in several of the other
   topologies it outperformed the Dijkstra algorithm for networks of up
   to about 16,000 nodes.  It should be noted that in those experiments
   no upper bound on the number of hops in a shortest path was set.




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   algorithm with Q quantized bandwidth values, the corresponding worst
   case asymptotic complexity is O(Q   .    (M logN)).  Both approaches
   provide solutions of comparable orders of complexity, whose exact
   merits depend on the respective values of H, Q and N.  If on-demand
   computations of QoS paths are practical, then a standard Dijkstra
   algorithm provides a solution of complexity O(MlogN).


G. Extension:  Handling Propagation Delays

   In general, the framework proposed for path selection does not allow
   us to explicitly account for link propagation delays.  As mentioned,
   this aspect is dealt with through a policy mechanism, which for
   delay-sensitive connections deletes from the topology database links
   with high propagation delays, such as satellite links.  However, it
   is worth pointing out that a simple extension to the proposed path
   selection algorithm allows us to directly account for delay in a
   number of special cases.  We proceed to describe next this extension
   and the case where it applies.

   A common way to map delay guarantees into bandwidth guarantees
   (e.g., consistent with the schedulers and corresponding delay
   bounds presented in [GGPS96, PG94]) is according to the following
   expression:


                     D(p) =A(h(p))=b +sum(l in p) d(l)               (1)

   where p is the path traversed, D(p) is the guaranteed (upper-bound)
   end to end delay, h(p) is the number of hops, b is the reserved
   bandwidth, d(l) is the (fixed) propagation delay of a link l, and A(h)
   is a parameter that grows with h (a typical value is A(h)= B +h . c,
   where B is the burst size and c is the maximum packet size).

   Since we deal with intra-domain routing, and since links with
   prohibitively high propagation delays are assumed to be filtered out
   by means of policy, it can be assumed that typically there is some
   value d which is a reasonable upper bound on the propagation delays
   d(l) of all links.  Expression (1) then implies that an end to end
   delay requirement D can be translated into a bandwidth requirement
   b(h) by the following expression:


                           b(h) =A(h)=(D -h. d)                      (2)


   where h is the number of hops on the path established for the
   connection.



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H. Accounting for Link Metric Inaccuracy in Path Selection

   Suppose that each node sends a Link State Advertisement (LSA) only
   when the ratio between the current value bw of a link and the last
   reported value is above (or below) some threshold, say 2. (10).
   This implies that, when a path with some b units of bandwidth is
   sought, links with advertised bandwidth values above 2 . b are ``safe
   bets'', those with values below b_2should be excluded, and all the
   rest may supply the required bandwidth with various degrees of
   certainty.  This means that a third objective is added to our two
   standard objectives of bandwidth and hop count, namely certainty.
   Its incorporation in the path selection process can be handled with
   various degrees of complexity and sophistication, of which we outline
   a few.  For further enhancements of these schemes the reader is
   referred to [GOW97].

(1) A probabilistic approach:  The bandwidth value of a link l is, for
   the decision maker, a random variable that takes values in (bl_2;2 .  bl),
   where blis the last advertised value.  Making some assumptions
   on the probability distribution of these values, e.g.  uniform
   distributions, one can compute for each bandwidth requirement b
   the success probability of a link l, say pl(b), and then run a BF
   algorithm on the metric {wl}, where wl  =  -log(pl(b)) (see [GO97] for
   details).  However, the problem here is that a different path should
   be computed for each bandwidth value b, hence rendering this approach
   too complex in the case of pre-computed routes.  We are thus behooved
   to consider a simpler approach.

(2) A simple approach:

   Here we run the standard BF algorithm, described in Section
   2.3.1, obtaining as an output an N  .  H  QoS routing table.  Let ff,
   0:5   ff    1, be a parameter that indicates the ``risk proneness''
   of the decision maker (the lower the value, the higher the risk
   proneness is).  Also, let HR be a parameter that indicates how
   many hops the decision maker is willing to trade for safety.  Then,
   upon accommodating a connection request with b values of bandwidth,
   perform the following:

    1. From the routing table, get hmin, the minimal number of hops of a
       path with bandwidth of at least ff.b units.

----------------------------
10. To keep the discussion simple, we do not bother here about potential
   oscillations when the values become very small, an issue that can be
   addressed by switching to an additive rule for such values.





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    2. From the routing table, get hmax, the minimal number of hops of a
       safe path, i.e., with bandwidth of at least 2.b.

    3. If hmin+ HR hmax:  choose the safe path.

    4. Otherwise:  from the table, choose the path that has the maximal
       bandwidth among those having at most hmin+ HR hops.


References

   [Alm92]  P. Almquist.  Type of Service in the Internet Protocol
            Suite.  INTERNET-RFC, July 1992.

   [BG92]   D. Bertsekas and R. G. Gallager.  Data Networks.  Prentice
            Hall, Englewood Cliffs, NJ, 2nd edition, 1992.

   [Car79]  B. Carre.  Graphs and Networks.  Oxford University Press,
            ISBN 0-19-859622-7, Oxford, GB, 1979.

   [CGR94]  B. V. Cherkassky, A. V. Goldberg, and T. Radzik.  Shortest
            Paths Algorithms:  Theory and Experimental Evaluation.
            In Proceedings of the 5th Annual ACM SIAM Symposium on
            Discrete Algorithms, pages 516--525, Arlington, VA, January
            1994.

   [CLR90]  T. H. Cormen, C. E. Leiserson, and R. L. Rivest.
            Introduction to Algorithms.  MIT Press, Cambridge, MA,
            1990.

   [GGPS96] L. Georgiadis, R. Guerin, V. Peris, and K. N. Sivarajan.
            Efficient Network QoS Provisioning Based on per Node
            Traffic Shaping.  IEEE/ACM Transactions on Networking,
            4(4):482--501, August 1996.

   [GJ79]   M.R. Garey and D.S. Johnson.  Computers and Intractability.
            Freeman, San Francisco, 1979.

   [GKH97]  R. Guerin, S. Kamat, and S. Herzog.  QoS Path Management
            with RSVP.  In Proceedings of the 2nd IEEE Global Internet
            Mini-Conference, Phoenix, AZ, November 1997.

   [GKR97]  R. Guerin, S. Kamat, and E. Rosen.  An extended rsvp
            routing interface.  INTERNET-DRAFT, June 1997.  work in
            progress.






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   [GLG+97] Der-Hwa Gan, T. Li, R. Guerin, E. Rosen, and S. Kamat.
            Setting up reservations on explicit paths using rsvp.
            INTERNET-DRAFT, December 1997.  work in progress.

   [GO97]   R. Guerin and A. Orda.  QoS-Based Routing in Networks with
            Inaccurate Information:  Theory and Algorithms.  In IEEE
            INFOCOM'97, pages 75--83, Kobe, Japan, April 1997.

   [GOW97]  R. Guerin, A. Orda, and D. Williams.  QoS Routing
            Mechanisms and OSPF Extensions.  In Proceedings of the 2nd
            IEEE Global Internet Mini-Conference, Phoenix, AZ, November
            1997.

   [Moy94]  J. Moy.  OSPF Version 2 - RFC No. 1583.  INTERNET-RFC,
            March 1994.

   [Moy97]  J. Moy.  OSPF Version 2 - RFC No. 2178.  INTERNET-RFC, July
            1997.

   [PG94]   A. K. Parekh and R. G. Gallager.  A Generalized Processor
            Sharing Approach to Flow Control in Integrated Services
            Networks:  the Multiple Node Case.  IEEE/ACM Transactions
            on Networking, 2:137--150, 1994.

   [Prz95]  A. Przygienda.  Link State Routing with QoS in ATM
            LANs.  Ph.D. Thesis Nr. 11051, Swiss Federal Institute of
            Technology, April 1995.

   [RZB+97] R. Braden (Ed.), L. Zhang, S. Berson, S. Herzog, and
            S. Jamin.  Resource reSerVation Protocol (RSVP) version 1,
            functional specification.  INTERNET-DRAFT, June 1997.  work
            in progress.

   [ST83]   D.D. Sleator and R.E. Tarjan.  A Data Structure for Dynamic
            Trees.  Journal of Computer Systems, 26, 1983.

   [Tan89]  A. Tannenbaum.  Computer Networks.  Addisson Wesley, 1989.



Authors' Address


   Roch Guerin
   IBM T.J. Watson Research Center
   P.O. Box 704
   Yorktown Heights, NY 10598
   guerin@watson.ibm.com



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   VOICE   +1 914 784-7038
   FAX     +1 914 784-6205

   Sanjay Kamat
   IBM T.J. Watson Research Center
   P.O. Box 704
   Yorktown Heights, NY 10598
   sanjay@watson.ibm.com
   VOICE   +1 914 784-7402
   FAX     +1 914 784-6205

   Ariel Orda
   Dept. Electrical Engineering
   Technion - I.I.T
   Haifa, 32000 - ISRAEL
   ariel@ee.technion.ac.il
   VOICE   +011 972-4-8294646
   FAX     +011 972-4-8323041

   Tony Przygienda
   Bell Labs, Lucent Technologies
   prz@dnrc.bell-labs.com
   VOICE   +1 732 949-5936

   Doug Williams
   IBM T.J. Watson Research Center
   P.O. Box 704
   Yorktown Heights, NY 10598
   dougw@watson.ibm.com
   VOICE   +1 914 784-5047
   FAX     +1 914 784-6318




















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