Weak Convergence Limits for Sojourn Times in Cyclic Queues Under Heavy Traffic Conditions

Abstract

We consider sequences of closed cycles of exponential single-server nodes with a single bottleneck. We study the cycle time and the successive sojourn times of a customer when the population sizes go to infinity. Starting from old results on the mean cycle times under heavy traffic conditions, we prove a central limit theorem for the cycle time distribution. This result is then utilised to prove a weak convergence characteristic of the vector of a customer's successive sojourn times during a cycle for a sequence of networks with population sizes going to infinity. The limiting picture is a composition of a central limit theorem for the bottleneck node and an exponential limit for the unscaled sequences of sojourn times for the nonbottleneck nodes.

Weak Convergence Limits for Closed Cyclic Networks of Queues with Multiple Bottleneck Nodes

“…This result generalizes the central limit theorem for the cycle time distribution when the number of nodes is fixed and all service rates are distinct [DMS08]. In Section 3 we analyze differences in the filling behaviour of the network in case of a single bottleneck and in case of multiple bottlenecks.…”

“…The usual interpretation for a cyclic network is that, with an increasing number of customers, the bottleneck approaches a Poissonian source feeding the rest of the network, while all the other nodes eventually form an open ergodic tandem system, the behaviour of which is well understood: local geometrical queue length distribution and independence over the nodes in steady state. A similar property holds for sojourn times and partial cycle times; see Theorem 5.1 of [10].…”

“…It is rather obvious that a similar property should follow for the non-bottleneck nodes in the one-bottleneck case from (1.3), resp., (1.4), for the sojourn times and partial cycle times and their asymptotic behaviour, see Theorem 5.1 in [DMS08].…”

“…Starting with the limiting cycle time distribution, we prove a central limit theorem for N → ∞, which generalizes the central limit theorem for the cycle time when all service rates are distinct [10]. In Section 4 we analyze the differences in the filling behaviour of the network between the single bottleneck and multiple bottleneck cases.…”

The cyclic queue and the tandem queue

“…Another way to overcome the lack of explicit results on sojourn time distributions is to use heavy traffic limiting results. In the closed cyclic queue this means that bottleneck analysis is performed, which is even in productform networks of value, due to computational problems when large populations are considered, see [6][Section II .7], and more recently [16]. For non exponential service times, see [6][Section III.7], and more recently [23].…”