On the waiting times in queues with dependency between interarrival and service times

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Abstract

In this paper a queueing system with partial correlation is considered. We assume that the amount of work (service time) brought in by a customer and the subsequent interarrival time are dependent. We show that in this case stronger dependence between interarrival and service times leads to decreasing waiting times in the increasing convex ordering sense. This generalizes a result of Chao (Oper. Res. Lett. 17 (1995) 47–51).

Introduction

Most queueing models considered in the literature assume independence between the service times and the interarrival times. In practice, however, they will be dependent. A typical example of such a situation is the case, when the arrival of a customer with a long service time discourages the next arrival. Therefore, the aim of this paper is to investigate the effect of dependencies between the service times and the subsequent interarrival times on the waiting times of the customers.

There are many articles on state-dependent queues, but most of them assume that the service and/or interarrival times depend on the queue length. There are only a few papers, which assume a direct dependency between arrival and service patterns. Hadidi [4], [5] considers a single-server queue, where the joint distribution of the interarrival and service times are from the class of the so-called Wicksell–Kibble bivariate exponential distributions. For this case he derives a recursion for the Laplace transform of the waiting time distribution. Mitchell and Paulson [8] present some simulation studies that indicate that the waiting times decrease monotonically with the correlation between service and arrival times. Chao [3] has shown this theoretically for the case of the class of bivariate exponential distributions of Marshall–Olkin type. It is the purpose of this paper to extend his result to single-server queues with arbitrary joint distributions of the service times and the subsequent interarrival times. Our main result will be, that stronger dependence between the arrival and service patterns (in the sense of being more positive quadrant dependent) leads to shorter waiting times in the increasing convex ordering sense.

Our paper is organized as follows. In the next section we will collect the most important definitions and facts about stochastic order relations. They will then be used in Section 3 to proof the main result. Finally, we will give some examples in Section 4.

Section snippets

Stochastic orders and dependence

The most important notion for positive dependence of bivariate distributions is the so-called positive quadrant dependence (PQD). We say that a bivariate random vector X=(X1,X2) is PQD, ifP(X1>a1,X2>a2)⩾P(X1>a1)P(X2>a2)∀a1,a2R,or equivalently, ifP(X1⩽a1,X2⩽a2)⩾P(X1⩽a1)P(X2⩽a2)∀a1,a2R.This positive dependence concept compares a bivariate distribution with a bivariate random vector of independent random variables with the same marginals. This can naturally be generalized to a dependence

Main result

We consider a single server queueing system. The interarrival time between customer n and n+1 will be denoted by Tn, and Sn shall be the service time of customer n. The bivariate random vector (Sn,Tn) may have an arbitrary bivariate distribution. We only assume that the vectors (Sn,Tn),n∈N are independent and identically distributed, i.e. there is only dependence between the service time of a customer and the interarrival time of the following customer, but no dependence between interarrival

Examples

1. Chao [3] considered the case of the bivariate exponential distribution of Marshall–Olkin type. His main result is a special case of Theorem 5. In fact, let (S,T) have such a bivariate exponential distribution with S∼exp(λ) and T∼exp(μ). Then the survival function is given byF̄(s,t)=P(S>s,T>t)=exp(−(λ−γ)s−(μ−γ)t−γmax{s,t}),where 0⩽γ⩽min{λ,μ} is a parameter for the degree of dependency between S and T.

It is easy to construct random variables S,T with this distribution from three independent

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