Discrete Time Stochastic Networks

In this section we study the relation between throughput and queue-length distributions in discrete-time closed cyclic networks when the number of customers and the number of nodes grows unboundedly. To state our findings more precisely: For each positive integer N , consider a cyclic network with n(N) nodes and k(N) customers. We suppose that there are only finitely many types of nodes. That is, at each node the service rate is one of only a finite number L (independent of N) of possible service rates. Assume that n(N) →∞ as N →∞ and also that the limiting customer density α := limN→∞ k(N) · n(N)−1 exists. We further assume that the limiting proportion of nodes of each of the L types exists as N → ∞. We show that the asymptotic throughput exists and can be computed as a function of the limiting customer density and the limiting proportion of nodes of each type. Also, we calculate the asymptotic queue length distribution as a function of the asymptotic throughput. It turns out that the asymptotic behavior of the system depends strongly on the limiting customer density as clearly would be expected. It is a consequence of our results that the following different cases may occur: (1) In case of low traffic, i.e., α = 0, the density of customers in the system eventually is so sparse that the asymptotic throughput is zero. Note that this occurs even though the population size k(N) can grow without bound. (2) In case of heavy traffic, with α = ∞, it is natural to expect bottlenecks to occur at those servers which have the slowest limiting service rates (as the queue length approaches infinity). Indeed we shall conclude that asymptotically-infinite mean queue lengths occur at precisely such servers. (3) It is more interesting that even if the limiting customer density is finite, there may be servers with asymptotically-infinite mean queue length. However, such behavior will occur only when the customer density is sufficiently large and only for those servers with the slowest limiting service rates.