DELAY SENSITIVE ROUTING IN PNNI-BASED ATM NETWORKS Dimitris Logothetis Malathi Veeraraghavan Data Networking Systems Bell Laboratories Lucent Technologies Lucent Technologies 200 Laurel Ave. Rm. 3D-429 101 Crawfords Corner Rd., Rm. 4F-509 Middletown NJ 07748 Holmdel NJ 07733
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Abstract In this paper we propose an enhanced path selection algorithm for ATM networks that implement the Private Network-Node Interface (PNNI) v1.0 standard. Our enhancement stems from the fact that we consider delay in determining the appropriate route for an ATM call. The paper presents two new path selection algorithms based on two di erent delay accumulation methods, namely the simple (additive) method, and the asymptotic method. The delay based path selection algorithm is achieved by adopting a constrained shortest path algorithm which optimizes two metrics simultaneously in contrast to Dijkstra's algorithm that optimizes only one. The execution times of these two algorithms are also compared and quanti ed based on the results of an experimental study. Using an analytical model we demonstrate the performance gain by adopting a delay-based routing procedure in simple network con gurations in terms of number of crankbacks. Our network con gurations and trac loading patterns showed that up to 36% less crankbacks can be obtained when adopting a delay-sensitive routing scheme as opposed to a simple shortest-path algorithm.
Subject Area: Broadband Switching Systems and ATM Networks (Network Routing)
1 Introduction An important property of integrated networks is their ability to support services with di erent Quality of Service (QoS) requirements, some of them with stringent delay requirements. Voice and real-time video trac are some applications that are delay sensitive that will constitute a signi cant portion of integrated network trac. When selecting a path for the call
one should consider the most appropriate route that will meet the QoS of the call while also maximizing the probability that the call will be accepted by all switches on the path. The Private Network Node Interface (PNNI) standard employs source routing for its path selection, i.e., the rst switch for a given connection is responsible to select an optimal route for the connection. In certain cases, the path that was selected by the source switch may not have enough resources to accommodate the call even though the rst switch selected that path. This may occur if the source switch has \old" information or due to hierarchical aggregation. In such cases, a procedure called crankback will occur where a connection that cannot be accommodated at a switch because of CAC failure will be cranked back to the source switch for an alternate route determination. This is obviously an undesirable action since it delays the connection setup procedure. The PNNI routing protocol is a link-state based scheme where ATM switches exchange updates with neighboring switches on the status of links and resource availability. The PNNI routing protocol has several hooks for disseminating QoS related parameters. These include peak to peak cell delay variation (p-pCDV) and maximum cell transfer delay (maxCTD), two delay associated link-state parameters that are advertised. Other link-state parameters include cell loss ratio (CLR), available cell rate (AvCR), and administrative weight (AW), a measure of an administrator's preference for a given link. All the above mentioned parameters are available for path selection. The AW, CLR and AvCR metrics can be easily combined to form path selection algorithms that are extensions of well-known shortest path algorithms such as Dijkstra and Bellman-Ford with the AWs used as link weights. However, if delay is considered in the path selection process the problem be-
comes a more dicult constrained shortest path problem. To motivate the need for delay-sensitive routing, consider, for example, the simple three node network of Figure 1. Assume that all links have the same value for the metric of \cost" (or administrative weight according to the PNNI notation) (i.e., w1;2 = w1;3 = w2;3 = w. Now let CDV1;2 = 100 msecs, and CDV1;3 = CDV2;3 = 10 msecs. If a routing policy is used that is based on shortest paths (or smallest cost paths) only which does not take delay into account, then all calls from node 1 to node 2 will rst be routed through link f1; 2g. Local call admission control at switch 1 will reject these calls and the call will be cranked back, if there is an end-to-end CDV requirement of anything less than 100 msecs. A delay sensitive routing scheme would have selected the longer route f1; 2g; f2; 3g in the rst place (assuming the CDV requirement is between 20 and 100 msecs) avoiding the crankback. We will use this simple network later to quantify the number of crankbacks experienced if delay-sensitive routing is not used. For this reason, we develop an analytical model to quantify the number of additional crankbacks in a network that runs a simple shortest path algorithm with AW as the link cost. Another dimension to the problem is how delay is accumulated along a path. Previous studies [1] have shown that cell delay variation (p-pCDV) and maximum cell transfer delay (maxCTD) are not necessarily additive, i.e., the delay through a path is not necessarily the sum of the delays of the individual links that the path is traversing. In Section 5 we present two accumulation methods, namely the simple additive method and a non-additive method, called the asymptotic method, de ned in the ATM Forum Traf c Management speci cation. Based on these two delay accumulation methods, we present two path selection algorithms that incorporate delay based on dynamic programming implementations of the constrained-shortest path problem. These algorithms are known to be pseudopolynomial in time complexity for bounded delay or AW constraints. The rest of this paper is organized as follows: In Section 2, we survey prior work while in Section 3, we provide an overview of the connection setup procedure as speci ed in the PNNI v1.0 standard. Section 4 presents our analytical model that quanti es crankback in a simple three node network. Section 5 presents the two delay accumulation methods and Section 6 presents our two delay-based path selection algorithms. Finally, Section 7 summarizes this paper.
2 Prior Work The problem of QoS-based routing has received a lot of attention recently. Ma and Steenkiste [2] reduce the multiple constraint routing problem to a single metric shortest path routing problem for leakybucket regulated, Variable Bit Rate (VBR) sources and Weighted Fair Queueing (WFQ) type of scheduling policies. It is our belief, however, that a QoS routing algorithm should not depend on any scheduling policy since scheduling is implementation speci c. Furthermore, the VBR leaky bucket regulated model needs to be modi ed to address QoS routing of CBR calls. Wang and Crowcroft [3] classify metrics as additive (d1;2 = d1 + d2), multiplicative (d1;2 = d1 d2 ) and concave (d1;2 = minfd1 ; d2 g) and show that polynomially bounded routing algorithms can be obtained when considering at most one additive metric or one multiplicative metric. When considering typical routing metrics such as delay, delay jitter, cost, loss probability and bandwidth they conclude that only one of delay, delay jitter, cost and loss probability can be used in conjunction with bandwidth to obtain a polynomially bounded routing algorithm. In the context of PNNI routing, loss probability is de ned as a convex metric (i.e., (d1;2 = maxfd1 ; d2 g)) and the above statement should be modi ed as \only one of delay, delay jitter, and cost can be incorporated in a polynomially bounded algorithm." Guerin and Orda [4] address the issue of QoS routing with inaccurate information in the context of PNNI-based ATM networks. Salama, Reeves and Viniotis [5] propose a distributed heuristic algorithm for delay constrained shortest path routing. Their algorithm, although polynomially bounded, does not always nd the optimal path. The Internet Engineering Task Force (IETF) has also addressed QoS-based routing. In [6] the issues of QoS routing are addressed and some possible solutions are discussed. Guerin et al. [7] propose the use of a variant of Bellman Ford's algorithm to compute paths of maximum available bandwidth for all hop counts as an improvement of existing routing schemes in IP networks which do not consider bandwidth. However, this scheme does not include a consideration of delay metrics. In our opinion, the previous approaches do not address the QoS routing problem e ectively since they either ignore delay altogether, or impose relationships between (available) bandwidth and delay metrics in order to convert a multi-metric optimization problem to a single metric shortest path problem. In this paper we address delay-sensitive routing in ATM networks that implement the Private Network-Network Interface (PNNI) standards [8] by explicitly considering delay and propose a solution based on the constrained shortest path algorithm. We also provide
experimental results listing execution times of the algorithm as well as analytical models to quantify the performance of the algorithm.
3 Background In this section we review the path selection process described in the PNNI standard. PNNI is a hierarchical, link-state routing protocol that organizes switching elements into logical groups called peer groups. Switches that belong to the same peer group are assigned a common Peer Group Identi er (PGID). Nodes that belong to the same peer group become aware of each other via the exchange of Hello packets. Nodes exchange database information using PNNI Topology State Elements (PTSE)s that contain topology characteristics derived from link or node state parameter information. PTSEs are grouped to form PNNI Topology State Packets (PTSPs) and PTSPs are ooded in each peer group and each node in the same peer group will have identical databases. Each peer group has a peer group leader, which is responsible for aggregating information and distributing aggregate information at the higher layers. Call establishment in PNNI consists of two operations; path selection and connection state setup at each point along that path. PNNI uses source routing for all connections setup requests. The path is encoded as a stack of Designated Transit Lists (DTLs), which is explicitly included in the setup signaling message. The source node will select a route based on topology, loading and reachability information in its database. Parameters carried in PTSPs include nonadditive attributes such as CLR and AvCR and (additive) metrics such as administrative weight, cell delay variation and maximum cell transfer delay. PNNI speci es that attributes can be used to prune the network graph during path selection. This process, which is polynomially bounded by O(l), where l is the number of links in the peer group and O() the standard asymptotic complexity notation. The PNNI standard does not specify how to use metrics to obtain the optimal route which is left to various switch vendors to implement di erent path selection algorithms. In Appendix H of the PNNI standard, Dijkstra's algorithm is mentioned for computing minimum cost routes based on link metrics. A scaled average of administrative weight and delay is also mentioned in Appendix H as a possible solution for route precomputation. Note however that this approach will not always produce a route that meets the delay requirement resulting in crankbacks. A similar situation arises if one chooses to optimize only the AW and not the delay. This is particularly true in cases where the AW is set independently of link congestion
and re ects some \a priori preference" of the network administrator for particular links/routes. It is easy to see that attributes can be easily incorporated in the path selection process when searching for the optimal shortest path route. Attributes are basically convex or concave \metrics" and a simple network pruning process will identify which links are not eligible for the path selection. The existence of more than one metric in the path selection process however, leads to a multiple constraint-shortest path problem which is known to be an NP complete problem [9]. This is probably why most previous e orts on QoS routing attempt to reduce the multiple metric optimization problem to a single metric problem by exploiting relations among metrics. It is our opinion, however, that since these relationships are valid under certain scheduling policies, such as weighted fair queueing and require certain input trac patterns, such as leaky bucket conforming trac, they cannot be part of a realistic multivendor network environment, where many switches with di erent bu er management and scheduling policies are present. In other words, a realistic path selection algorithm should not attempt to exploit relationships among advertized parameters, unless the assumptions for these relationships are based on standardized architectures and principles. We therefore believe that it is inevitable to optimize more than one metric simultaneously. We will be consequently facing a more complex optimization with no \exact" polynomial time solution as opposed to single metric shortest path algorithms that can be solved in polynomial time. In order to tackle the multiple metric optimization problem, let us make the following observations. First, we observe that delays and administrative weights typically only assume integer values. Also even if the theoretical range of these values is large (32 bits are allocated in the PTSPs for each metric) the practical range of values of these quantities is much smaller. Since maxCTD and CDV are related to each other, it may also be possible to consider only one of the two delay related metrics, CDV or maxCTD. For example, in a WAN environment maxCTD dominates the delay component whereas in the LAN environment CDV and maxCTD will be close to each other. We can therefore reduce the three metric optimization problem to a two metric optimization problem, namely one that considers AW and CDV, or one that considers AW and maxCTD. This problem is known as constrained shortest path problem in graph theory and there exist pseudo polynomial time algorithms for its solution. Indeed, as we shall see later it is possible to de ne dynamic programming procedures that solve the constrained shortest path in pseudopolynomial time provided that at least one of two metrics is bounded.
Another interesting observation regarding metrics is that metrics do not have to be additive when computing shortest paths. As we shall see later only a \dominance" property is necessary for this computation, i.e., if Di;j , Dj;k , Di;j;k denote the values of a metric on link (i; j ), link (j; k) and path (i; j; k), respectively,
Di;j;k = Di;j Dj;k > Di;j Di;j;k = Di;j Dj;k > Dj;k
(1)
We will also specify pseudopolynomial constrained shortest path algorithms for non-additive delay metrics.
4 The analytical model In this section we describe an analytical model to motivate the need to consider delay based metrics while selecting shortest paths. We initially focus on the three-node network of Figure 1 and consider only trac that originates from node 1 and destined to node 2. There are two disjoint paths that connect these two nodes; a direct one-link path and an alternate two-link path through node 3. We assume that call arrivals follow a Poisson process with parameter while call durations are exponentially distributed with parameter . We also assume that the bandwidth requirement of each call at each link is 1 unit of capacity. All links have the same capacity of C units. The measure of interest is the crankback ratio, the ratio of the number of calls that were cranked back, given that they could be accommodated on the network over the total number of calls. For analytical tractability we ignore the time associated with the routing decision process. We also ignore the time associated with the propagation of PTSP messages to the switch that performs the routing decisions. We assume that the administrative weights are xed (i.e., they do not change over time as the network loading changes) and they are all equal to one (i.e., w12 = w13 = w23 = 1). Then all calls will be rst attempted to be routed to the shortest path i.e., the path that connects directly the nodes 1 and 2. As more and more calls are routed through the direct link (1,2) the link's delay will increase and it may be possible that the delay requirement of the call is violated, even if there is still enough available capacity to accommodate the call on the direct link. In such a situation a crankback will occur and on the second attempt the alternate route will adopted. Let Kmax, Lmax denote the maximum number of calls that can be accommodated on the direct and alternate route, respectively, without violating the endto-end CDVc requirement for the call. Determining
the relationship between Kmax, Lmax and the available capacity, AvCR is a very dicult task, since it depends on many unknown factors such as switch architecture (bu ering scheme and scheduling policy), link speed, link capacity partitioning among service categories and trac patterns. As a general trend, however, it is expected that, in most cases, the more stringent the end-to-end CDVc , or the smaller the AvCR the smaller Kmax and Lmax. For our numerical results we obtain crankback ratios as a function of Kmax and Lmax since a relationship of the form Kmax = f (CDVc ; AvCR) is not possible. The only assumption that we make for our model is that Kmax and Lmax are smaller than C or in other words delay is the binding constraint. De ning the two-dimensional stochastic process f(X (t); Y (t)); t 0g with X (t) the number of calls on link (1,2) at time t and Y (t) the number of calls on the route that connects links (1,3) and (3,2), it is easy to see that this discrete state continuous time stochastic process is a Continuous Time Markov Chain (CTMC) and its state diagram is depicted in 2. Let us also de ne Nc(t), the number of calls that experience crankback up to time t, given the calls can be accommodated, and Na (t), the number of attempted calls up to time t. We are interested in the following measure that quanti es crankback:
E [Nc (t)] f = tlim !1 E [N (t)] a
(2)
A transition from state (Kmax ; i) to (Kmax; i + 1), i = 0; 1; ; L ? 2 will lead to a crankback due to a CDV requirement violation even though there is enough bandwidth on the direct link. Associating a reward of value 1 with these state transitions and using the theory of Markov Reward Models [10] it can be shown that: E [Nc (t)] f = tlim !1 E [Na (t)] =
LX ?1 i=0
K;i
(3)
where i;j is the steady-state probability of the CTMC of Figure 2 and are given as the solution of the Equation: Q = 0: (4) Obtaining i;j in closed-form for arbitrary Kmax, Lmax , where Lmax is the maximum number of calls that can be accommodated on the alternate path via node 3 before the CDV requirement is violated, is not an easy task. We therefore rely on numerical methods, i.e., we solve numerically the linear system of the underlying CTMC.
Tables 1 and 2 show crankback ratios for values of equal to = 0:2calls=sec and = 10calls=sec, respectively, as a function of Kmax and Lmax. We also assume = 1call=sec. The results are obtained from the numerical solution of the CTMC. The Erlang B formula which is given by: K Erl(K; a) = PaK =Ki ! (5) i=0 a =i! is also shown for comparison. Note the remarkable accuracy of the Erlang B formula for light loads and large values of Kmax. The accuracy of the Erlang B formula motivates us to seek for approximate expressions for our crankback measure in larger and more general networks. Let us generalize the network of Figure 1 to a M disjoint path network of Figure 3. We assume that AWi AWj for i < j , where i; j are path indices. Under this assumption an incoming call is routed rst to the lowest indexed path, p1 . If this path is unavailable or heavily loaded then the call will be routed to the next smallest indexed path. A shortest-path algorithm that optimizes the administrative weight will always attempt to route on p1 path rst. If the call is cranked back, the next (higher index) path is considered. An approximate expression for the crankback ratio will be:
fk =
MX ?1 l=1
0l 1 X l Erl @ Nj ; A j =1
(6)
assuming that the source node (node 1) is capable of recognizing all routes to node 2 and that whenever the rst l routes do not meet the CDV test (i.e., CDVc < CDVe(?l)e 8l) l crankbacks will occur before selecting the (l + 1)th path. A numerical example illustrating the above formula is now shown. We use the same values as before for connection parameters and furthermore all alternate routes go through an intermediate switch (i.e., all alternate routes consist of two switches). Assuming the additive delay accumulation method we obtain N1 and N2 = N3 = = NM as before. The results are summarized in Table 3 for M = 10 routes. The numbers presented in the tables above suggest (for the considered network con gurations and traf c loading patterns) that up to 36% of the calls that could be accommodated on the alternate route would have to be cranked back whenever a simple shortest path route that considers only the administrative weight is used. It is important therefore to develop enhanced path selection algorithms that consider delay metrics also. In the next section we present a series of algorithms that incorporate delay in the routing process.
Node 3
Call with dest. Node 2 Node 1
Node 2
Figure 1: An example three-node network λ 0,0
1,0 µ
µ
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1,1 µ λ
Lµ 0,L
2µ
Kµ
2µ
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λ K,1
Kµ
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Lµ 1,L
µ
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0,1 µ
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Figure 2: The CTMC for the number of calls in the network Table 1: Crankback ratio for di erent values of CDVc , = 0:2 (underloaded regime) Crankback Ratio Erlang B 0.150273 0.166667 0.00104 0.00109 0.0000545 0.0000546 0.0000546 0.0000546 2:18 10?6 2:18 10?6 7:27 10?8 7:27 10?8 7:28 10?8 7:28 10?8 2:08 10?9 2:08 10?9
Kmax Lmax Num. Analysis 1 3 4 4 5 6 6 7
1 1 2 3 4 5 6 7
Table 2: Crankback ratio for di erent values of CDVc , = 10 (overloaded regime) Crankback Ratio
Kmax Lmax Num. Analysis Erlang B 1 3 4 4 5 6 6 7
1 1 2 3 4 5 6 7
0.0894188 0.0854 0.16215 0.2376 0.2907 0.3213 0.3648 0.3522
0.909091 0.732 0.6466 0.6466 0.5639 0.4845 0.4845 0.409
5 Delay Accumulation Methods We now present two di erent delay accumulation methods; the simple method and the asymptotic method.
5.1 Simple Method
The simple method for accumulating peak-to-peak CDV during call setup is the following: A switch receives a call setup message with the accumulated peak-to-peak CDV and it adds its contribution of peak-to-peak CDV ( ) to the accumulated peak-topeak CDV. This simple method, which is de ned in TM 4.0 [11], is based on estimating the end-to-end CDV ( ) as the sum of individual CDV ( ) values along the path from source to destination. If there are N switches along the path and if we denote the quantile of CDV in switch i by CDVi ( ), then the total accumulated CDV is:
CDVtotal ( ) =
N X i=1
CDVi ( )
(7)
The simple method requires only one parameter, CDV ( ), for its computation. The estimated CDV is always an upper bound of the actual CDV but it may be very conservative for connections that traverse many hops.
5.2 Asymptotic Method
The asymptotic method is described in Information Appendix V of [11]. It uses both the mean and the variance of transfer delay, and the actual delay variation in each switch, in order to compute the end-toend CDV. The error is compensated by adding the maximum di erence between the estimate and actual CDV in the switches along the path. The end-to-end CDV over N switches, assuming independent delays in the switches, is given by: CDV total ( ) =
N X
v u N uX i + t i2 t( )
i=1 + 1max fCDV i ( ) ? (i + i t( ))(8)g; iN i=1
where t( ) denotes the quantile of standard normal distribution, i denotes the mean delay in switch i, and i denotes the standard deviation of delay in switch i. The quantity di ( ) = fCDV i ( ) ? (i + i t( ))g (9) is referred to as discrepancy. This method also yields an upper bound of the actual CDV but the bound is
much tighter when compared with the simple method. The asymptotic method requires each switch to report four parameters: mean queueing delay (), variance of queueing delay (), discrepancy parameter d( ), and xed delay f . PNNI 1.0 does not support these parameters. Only maxCTD and CDV are supported.
6 Improved path selection algorithm In this section we present two methods to perform delay-based path selection. These methods correspond to the two CDV/CTD accumulation methods described in Section 5, namely, the additive method and the asymptotic method. We will assume that the rst switch in the peer group computing the source route assumes the CDV link reported for the service category of the type being set up to hold even after this new connection is accepted. Before proceeding to the description of the algorithms we clarify that we incorporate delay through only one parameter, either CDV or CTD. A multidimensional optimization is also possible but more complex. We will present the algorithms using CDV as the delay parameter. The CTD case is obvious.
6.1 Additive delay-based path selection algorithm
As mentioned in Section 2, PNNI v1.0 speci cation states that CDV and maxCTD are (additive) metrics and hence can be used in the shortest path computation together with administrative weights, AW. As mentioned earlier, a \shortest-path" algorithm that optimizes more than one metric (e.g., AW and CDV) simultaneously is an NP-complete problem and Dijkstra's algorithm cannot be used in this case [9]. Nevertheless, whenever the \aggregate metrics" (e.g., total delay or total AW) are bounded and take discrete integer values, pseudo-polynomial algorithms can be constructed for route generation. A rst approach suggested in Appendix H of PNNI v1.0 is to perform two independent \shortest path" runs, one using AW as the cost function and one using CDV/CTD. The obvious disadvantage of this approach comes from the fact that an \optimal path" according to one cost function may not necessarily be optimal according to another cost function. Another approach is to solve the constrained shortest path problem using dynamic programming procedures (see [12] and references therein) that assume a discretized and bounded domain for the CDV and perform an exhaustive search over it. This pseudo-
polynomial time algorithm for acyclic graphs is outlined below for the one source, all destinations case: De ne lij and tij to be the AW and the CDV of the link that connects nodes i and j , respectively. Let T be an upper bound on the end-to-end CDV of any path in the network and fi (t) the length of a shortest path from node 1 (the source) to node i with CDV less or equal to t. 1. Initialize f1 (t) = 0; t = 0; ; T 2. Initialize fj (0) = 1; j = 2; ; N 3. Compute fj (t) = min fj (t ? 1); minkjtkj t ffk (t ? tkj ) + lkj g j = 2; ; N , t = 1; ; T The complexity of the above algorithm is O(lT ), with l being the number of links in the network. Compare this with Dijkstra's algorithm which is O(n2 ), with n as the number of nodes in the network1. We now proceed to describe the steps required for an additive delay-based path selection using a constrained shortest path algorithm. The algorithm includes four steps:
1. Precompute \shortest path" routes a priori using the constrained shortest path algorithm described above. Store routes for di erent values of CDV (or t). For example if T = 25 msec, we may choose to compute routes for t1 = 15 msecs, t2 = 20 msecs and t3 = 25 msecs. If desirable, perform Generic Connection Admission Control (GCAC) (simple or complex) a priori at each link of the \network" assuming some xed trac descriptor to determine which links of the \network" have the necessary capacity to accommodate the connection. Store the path(s) in a cache of the form (Trac descriptor, CDV, route). 2. When a call setup request arrives at a switch, select an optimal path from the set of precomputed routes based on the values of the STD and QoS parameters in the call setup request. 3. If no precomputed path is found perform an ondemand route computation by rst performing capacity-based GCAC (simple or complex as described in the Appendix) to eliminate links that do not meet the cell rate requirements and then run the constrained shortest path for the given call requirement. If the constrained shortest path algorithm is considered prohibitively expensive (large and dense networks) then run Dijkstra's algorithm with CDV as the additive parameter. 1 We do not assume any special structure such as heaps (or priority queues) to reduce the time complexity.
This solution requires no changes to the existing standards.
6.2 Delay based path selection algorithm for the asymptotic method
As mentioned earlier the end-to-end CDV and CTD are, in general, not additive. We should therefore attempt to modify shortest path algorithms, (e.g., Dijkstra's algorithm and constrained shortest path algorithm) to allow for non-additive metrics. We rst show how to modify the dynamic programming procedure for the constrained shortest path algorithm presented in Section 6.1. De ne lij and tij to be the AW and the CDV of the link that connects nodes i and j , respectively. Let L be an upper bound on the end-to-end CDV of any path in the network. Let gi (l), i (l), i2 (l) and di (l) be the CDV, the (cumulative) mean of the delay, the (cumulative) variance of the delay and the discrepancy of the delay of a shortest path from node 1 (the source) to node i with AW at most l. 1. Initialize g1 (l) = 0; l = 0; ; L 2. Initialize gj (0) = 1; j = 2; ; N 3. Initialize j (l) = j2 (l) = dj (l) = 0, l = 0; ; L, j = 2; ; N 4. Compute gj (l) = min gj (l ? 1); minkjlkj l fgk (l ? lkj ) tkj g , j = 2; ; N , l = 1; ; L with gk tkj is de ned as: k (l ? lkj )
+ +
?p2 (l ? lkj ) + 2 t( )e k kj max dk (l ? lkj ); dkj ( ) kj + d
(10)
kj , kj2 and dkj ( ) denote the mean, the variance and the discrepancy of link that connects nodes k and j . t( ) is the quantile of the standard normal distribution, that, in general, has a non-integer value. Te execution time of the above algorithm is O(mL). The steps required to perform a delay-based path selection based on the asymptotic method are shown below: 1. Precompute \shortest path" routes a priori using the constrained shortest path algorithm described above. Store routes for di erent values of AW (or l). If desirable, perform GCAC (simple or complex as described in the PNNI v1.0 standard [8]) a priori at each link of the \network" assuming some xed trac descriptor to determine which links of the \network" have the necessary capacity to accommodate the connection. Store the path(s) in a cache of the form (Trac descriptor, CDV, route).
2. When a call setup request arrives at a switch, select an optimal path from the set of precomputed routes based on the values of the STD and QoS parameters in the call setup request. 3. If no precomputed path is found perform an ondemand route computation by rst performing capacity-based GCAC (simple or complex as described in the Appendix) to eliminate links that do not meet the cell rate requirements and then run the constrained shortest path for the given call requirement. If the constrained shortest path algorithm is considered prohibitively expensive (large and dense networks) then run a shortest path algorithm with CDV as the only parameter.
6.3 Execution times comparison
A quantitative comparison of the execution times of the above described algorithms as a function of the number of network nodes is shown for dense and sparse networks in Tables 4 and 5, respectively. The time and length intervals are taken to be T = 100 and L = 100 for the additive constrained shortest path and asymptotic method, respectively. A measure called \connectivity" was used while generating random topologies for this exercise. Link lengths were uniformly distributed in the interval (1; 3), while link delays are uniformly distributed in the interval (1; 10). From the tables above we observe that both the additive and asymptotic constrained shortest path algorithms are, as expected, consistently slower than Dijkstra's algorithm and do not scale well for large and dense networks. In addition, the constrained shortest path algorithm based on the asymptotic method is consistently slower than the one based on the additive method. We will therefore recommend the use of these algorithms only for \precomputations" rather than \on-the- y" computations. In summary, we propose to include CTD/CDV during route precomputation, and if required, in the ondemand route determination. The introduction of non-additive delay metrics will require support for new parameters in the Resource Availability Information Group (RAIG) for the asymptotic CDV computation. Additionally ecient computational algorithms need to be found that provide the optimal paths with respect to both the administrative weight and the CDV/CTD constraint.
7 Summary This paper proposed enhanced path selection algorithms that consider delay-related metrics while determining the route for a connection. Currently, in PNNI networks the Generic Connection Admis-
p
.. . p Call with dest. Node 2
M
3
p2 p
1
Node 1
Node 2
Figure 3: A M disjoint path network
Table 3: Approximate crankback ratio for di erent values of CDVc , = 1(critically loaded regime)
N1 3 4 4 5 6 6 7
Nodes 5 7 10 12 15 18 20
Ni i = 2; ; 10 Crankback Ratio 1 2 3 4 5 6 7
Links 7 15 31 45 73 108 141
Dijkstra's algorithm (secs) 14 25 51 70 111 150 188
0.104943 0.0164344 0.0155309 0.00306951 0.000511005 0.000510988 0.0000729927
CSP (add. del.) (secs) 759 1434 2671 3705 5635 8046 10505
CSP (Asym. meth.) (secs) 5107 9402 18036 26635 41101 59283 75614
Table 4: Exec. times for dense networks (conn. = 0.7)
Nodes 5 7 10 12 15 18 20
Links 2 2 7 11 17 32 31
Dijkstra's algorithm (secs) 13 25 51 71 109 153 183
CSP (Add. del.) (secs) 566 893 1726 2379 3892 5225 6038
CSP (Asym. meth.) (secs) 3221 4840 9791 14113 21216 32916 37356
Table 5: Exec. times for sparse networks (conn. = 0.2)
sion Control (GCAC) algorithm checks available cell rates on links, and the shortest path algorithm only uses administrative weights as link metrics. To enhance this solution, two variations of a delay-based shortest-path algorithm were proposed corresponding to the two delay accumulation methods described in the ATM Forum Trac Management Speci cation. The rst proposal assumes that Cell Delay Variation (CDV) is an additive metric. A constrained shortestpath algorithm is used to determine the shortest path with respect to administrative weights while meeting the end-to-end CDV. For the additive case, we implemented and compared the time complexity of the constrained shortest path algorithm and Dijkstra's algorithm for various network sizes and connectivity assumptions. Our results suggest that the constrained shortest path algorithm can be computationally expensive ( 1 second execution time that gives a number two orders of magnitude slower than the corresponding execution time using Dijkstra's algorithm) for large and dense networks and therefore should be avoided in the on-demand route computation phase. For sparse networks, the complexity is somewhat reduced. Note that Dijkstra's execution times are practically insensitive to the degree of connectivity of the network. The second proposal assumes that delay is not additive and is accumulated using the asymptotic method. The algorithms presented in this paper may also be applied to other QoS capable networks, such as the next generation of the Internet, that support the Resource Reservation Protocol (RSVP) and the Internet Protocol (IP)v6.
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