Delays in a series of queues with correlated service times

Abstract

In almost all applications of queueing network models it is assumed that for each customer the service times at different network nodes are independent. But service times in, for instance, computer and communication networks are typically essentially determined by properties like message or packet lengths that do not change substantially on the route through the network. Therefore, the service times of any customer in a queueing network are likely to be correlated, which can significantly influence quality of service (QoS) properties and performance measures such that results obtained with the independence assumption may be misleading. We consider delays in a series of queues with correlated service times at each network node where for each customer the service time at the first node is a random variable and the successive service times are correlated with the one at the first node. A recursive scheme for delays is provided. This scheme is used in order to efficiently conduct a simulation study where two types of correlation are studied, namely identical service times, and service times with an additional Gaussian noise. The simulation study focuses on comparisons of end-to-end delays for independent service times at different nodes and correlated service times, respectively. It turns out that for both correlation types, in light traffic the delays in case of correlated service times are larger than for independent service times by a factor that first increases with increasing traffic intensity up to a maximum value approached in medium traffic after which it decreases quickly and drops down to become significantly smaller than one in heavy traffic. This effect intensifies with increasing number of network nodes and depends, as well as the crossover point from which on correlated service times yield smaller delays, on the distribution of the service times at the first node.