Computational Techniques for Accurate Performance Evaluation of Multirate, Multihop Communication Networks Albert G. Greenbergy AT&T Laboratories
R. Srikantzx Coordinated Science Laboratory and Department of General Engineering University of Illinois Abstract
Computational techniques are presented for connection-level performance evaluation of communication networks, with stochastic multirate trac, state dependent admission control, alternate routing, and general topology | all characteristics of emerging integrated service networks. The techniques involve solutions of systems of xed point equations, which estimate equilibrium network behavior. Though similar techniques have been applied with success to single-rate fully connected networks, the curse of dimensionality arises when the techniques are extended to multirate, multihop networks, and the cost of solving the xed point equations exactly is exponential. This exponential barrier is skirted by exploiting, in particular, a close relationship with the network reliability problem, and by borrowing e ective heuristics from the reliability domain. A series of experiments are reported on, comparing the estimates from the new techniques to the results of discrete-event simulations.
1 Introduction Analytic xed point methods are presented for the analysis of blocking in multihop, multirate integrated service networks with state dependent admission control and alternate routing. The analysis is at the connection level, and applies to synchronous transfer mode (circuit-switched) services, and to asynchronous transfer mode (ATM) services where e ective bandwidths [14] are used for admission control and routing. The methods are ecient, and as indicated by the numerical experiments below, are accurate. In recent years, there has been much interest in xed point methods, also known as reduced load approximations, for network analysis [7, 8, 13,19,26]. Fixed point computations are an appealing alternative and complement to discrete-event simulations, especially in network design and in performance evaluation under overload conditions. In addition, xed point methods have tremendous parallelism, which can be simply and dramatically exploited on today's parallel computers [13]. However, in large measure, the work to date on xed point methods was motivated by today's large voice networks. Commercial voice networks di er in two important ways from emerging integrated service networks: In commercial voice networks, the topology is completely connected (or nearly so), and routing is restricted to at most two hops. That is, a call for a given node pair is routed either on the link directly connecting the pair or via at most one intermediate node. In integrated service networks and some military circuit-switched voice networks, topologies tend to be more sparse, and calls may be routed on longer paths (with lengths, say, in the range 1{6 hops). In voice networks, all calls are of the same bandwidth. In integrated service networks, the network must accommodate a wide range of call bandwidths. An earlier version of the paper appeared in the Proceeding of the ACM Sigmetrics, March 1995, Ottawa, y Room 2C-119, AT&T Laboratories, 600 Mountain Av., Murray Hill, NJ 07974;
[email protected] z 131 C&SRL, 1308 W. Main Street, Urbana, IL 61801;
[email protected] x This work was done when the second author was at AT&T Bell Laboratories
Canada
Unfortunately, the straightforward generalization of the xed point method for fully connected singlerate networks, to general multirate networks leads to a combinatorial explosion in the systems of equations. Indeed, the computational complexity is exponential in the number of routes between node pairs, and is exponential in the number of bandwidth classes. The computational complexity due to the di erent classes has been studied in literature [2,3,8,11]. In this paper, we present a new approach to the problem of dealing with overlapping alternate paths through approximations drawn from recent progress in the area of network reliability. We also show that our approach combined with the approximation suggested in [2,3] to deal with multiple classes with state-dependent arrival rate yields accurate estimates of the blocking performance of multirate, multihop networks. The paper is organized as follows. In Section 2, the network model is described. We consider simple admission and routing controls, which are known to be e ective in single-rate fully connected networks, and have been proposed for integrated services networks (e.g. [10]). Admission controls are of trunk-reservation type [19]. Associated with each node pair is an ordered list of paths. A call is routed on the rst path in the list where the call is admissible, or is blocked if the call is not admissible on any of the paths. In Section 3, we present the assumptions behind the xed point method, and then describe the method. In Section 3.1, an example illustrates a novel and fundamental relationship with the network reliability problem [9]. In Sections 3.2 and 3.3, we present the details of the two mappings that constitute the xed point method. One mapping (Section 3.2), which is the main contribution of this paper, poses independent path-restricted network reliability problems for each node pair. To solve these problems, we use a recent, ecient linear programmingmethod of Boros, Prekopa and Lih [5,6,21{23]. The second mapping (Section 3.3) poses independent blocking probability computations for each link. Here, we use algorithms of Kaufman and Roberts [18,25] and Bean, Gibbens and Zachary [2,3]. This combination of algorithms skirts the exponential barriers mentioned above. Section 4 reports on the results of extensive numerical experiments, comparing the call blocking estimates from the xed point method with results from discrete-event simulations. An interesting byproduct of our experiments on a network (derived from an existing commercial network) is that bistability that occurs in fully connected networks (see, for example [19]) with no admission control does not occur in general topology networks. We will discuss this more later. Work on xed point methods for network analysis dates back at least as far as the 1950s (cf. [27]). Essentially all prior work has addressed single-rate networks. An exception is the work of Chung and Ross [8]. Very early related work of Jacobaeus and Lee [16,20] (1950 and 1955, respectively) on multistage switching networks [15] brought out diculties in dealing with path overlap (Section 3). Jacobaeus' work incorporates use of the inclusion-exclusion formula for special structures. The inclusion-exclusion formula is the starting point for the work of Boros and Prekopa [5, 21{23]. To handle path overlap, Lee proposed an approximation based on an assumption of path independence. In our application, Lee's approximation leads to poor estimates. However, useful estimates can sometimes be obtained at low cost using a certain conditional path independence assumption (Section 3.3).
2 Model We consider general networks on N nodes, indexed 1, 2, ..., N. A link (i; j) between nodes i and j has capacity Ci;j , counted in bandwidth units termed trunks. Links are undirected; (i; j) and (j; i) denote the same link. Calls o ered to the network fall into S classes, di erentiated by bandwidth and possibly by priority via trunk reservation, as described below. A call of class s has bandwidth bs, meaning that if the call is admitted to the network then it is allocated bs trunks on each link of a path from its source to its destination. A link's state is characterized by the number of calls in progress ns of each call class. A call is admissible on a given path if it is admissible on every link of that path. Consider a call of class s o ered to node pair (i; j). Link admissibility may be decided with or without control. In the absence of control, P the call is admissible on a link if the call's bandwidth bs is less than or equal to the idle bandwidth C ? t btnt . Admission controls considered here are of trunk reservation type. Speci cally, thePcall is admissible on link (i; j) if the call's bandwidth bs is less than or equal to the idle bandwidth C ? t btnt 2
plus r, where r is a static parameter, which may depend on s, i, j, and other factors, such as the length of the route associated with the call. Thus, choosing parameter r to be small allows the call type greater access to scarce capacity. If admitted on a path in the network, the call simultaneously seizes for its exclusive use bs trunks on every link of the path. Similarly, the call simultaneously releases bs trunks on every link of the path when it completes. We consider routing policies where each node pair (i; j) is assigned an ordered list Pi;j of paths between (1) (2) nodes i and j: Pi;j , Pi;j , .... A call is admitted on the rst path in the list where it is admissible. If the call is not admissible on any path then it is blocked; that is, rejected and lost. Trac is modeled as follows. Calls of class s are assumed to arrive to node pair (i; j) according to a Poisson process with rate i;j (s). If admitted to the network, the call holds for an exponential period of time with mean i;j (s). Thus, these calls constitute a load of i;j (s) = i;j (s)=i;j (s). The key performance metrics are the blocking probabilities, Prfblocking a call of class s arriving to node pair (i; j)g; for each node pair and each call class, or (equivalently) the admissibility probabilities, de ned as corresponding probabilities of admitting the call.
3 Fixed Point Method The xed point method is built on two approximations: [A1 (Link Independence)] The probability that a call is admissible on a given path in the network is the product of the probabilities that the call is admissible on each link of that path. [A2 (Poisson Rates)] Arrivals of calls of a given class to a given link are described by a Poisson process. The rate of the process may depend on the state of the link. Though arrival rates to node pairs are given and are assumed to be Poisson, admission control and routing thins and superposes the original arrivals onto the links in a state-dependent way, perturbing their Poisson character. Assumptions A1 and A2 lead to a system of equations whose unknowns are, for each link (i; j) and call class s: arrival rates i;j (s) of calls of class s from any node pair that includes link (i; j) on some route, given that link (i; j) is in a state that admits calls of class s, and probabilities ai;j (s) that link (i; j) is in a state that admits calls of class s. As described in Section 3.2, A1 and A2 together provide an independent mapping for every node pair (i; j) from probabilities ak;l (s) (with k and l dependent on the routing structure) to rates i;j (s). As described in Section 3.3, A2 provides an independent mapping for each link (i; j) from rates i;j (s) to probabilities ai;j (s). To estimate equilibrium network performance, we solve for the xed point of the composition of the mappings: arrival rates ! admissibility probabilities ! arrival rates, simultaneously xing the i;j (s) and ai;j (s). In the absence of admission control, under certain conditions multiple xed points may exist, re ecting inherent bistable behavior in the network [19]. Though a theoretical explanation is lacking, experience to date with xed point methods indicates no uniqueness problems for realistic problems when admission control is used.
3.1 Call Blocking and Network Reliability
Of course, A1 is an assumption of statistical independence. To understand its impact consider the starlike network of Figure 1. To a completely connected core of N nodes is added a satellite group of N nodes, with the two groups in one to one correspondence. A node in the satellite group has just one link, connecting it to the corresponding node of the core group. Suppose that each link between core nodes is admissible with probability and that each link between core and satellite nodes is admissible with probability . By A1, it follows that a call between a pair of satellite nodes is admissible with probability ? 2 1 ? (1 ? )(1 ? 2)N ?2 (1) 3
Figure 1: 10 node starlike network, which consists of a completely connected group of 5 nodes and a satellite group of 5 nodes. because the call must be admissible on both of the two links between satellite and core nodes (probability 2 ) and on at least one of the N ? 1 paths through the core nodes (probability 1 ? (1 ? )(1 ? 2)N ?2 ). This admissibility computation is an instance of the two-terminal network reliability probability, which has received intensive study (cf. [9,24]). In this problem, the goal is to nd the probability that there exists a reliable path between two nodes (terminals) in a given graph, where each edge e fails independently with given probability p(e), which may depend on e. Thus, Bernoulli trials with general probabilities determine which edges fail. A path is reliable if it includes no failed edges. In our application, we face a path-restricted version of the problem, where we ask just for the probability of a reliable path among a list of given paths, rather than among all possible paths. In general the pattern of overlap can be much richer than in the example of Figure 1, and the counterpart of equation (1) much more complex. Unfortunately, the two-terminal network reliability problem is #Pcomplete [9]. In the path-restricted problem, even if the length of the list is only of the same order as the number of nodes, the problem remains at least NP-hard. In general, in the worst case no more ecient method is known for the path-restricted problem than to apply the inclusion-exclusion formula, as follows. Consider a node pair with K paths between the nodes. Let Ak denote the event that path k is reliable, i.e., all the links in the path are available. The probability that there is at least one reliable path is
R = Prob(A1 [ A2 [ : : : [ AK ): By the inclusion-exclusion principle,
R=
K X k=1
(2)
(?1)k?1Sk ;
where Sk , referred to as the kth binomial moment, is given by Sk =
X
l1
