The Correlation Structure of the Output Process of Some Single Server Queueing Systems

Abstract

A queueing system can be regarded as transforming one point process into another (as pointed out for example in Kendall (1964), Section 6), namely, the input or arrival process with inter-arrival intervals $\{T_n\}$ is acted on by a system comprised of a queue discipline and a service (or, delay) mechanism, producing the output or departure process with inter-departure intervals $\{D_n\}$. The object of this paper is to study the correlation structure of the sequence $\{D_n\}$ (and this sequence we shall for convenience call the output process of the system) when the input process is a renewal process and when the service times $\{S_n\}$ (assumed to be independently and identically distributed, and independent of the input process) are such that the system can and does exist in its stationary state. In particular, we shall be concerned with conditions under which the process $\{D_n\}$ is uncorrelated, by which we mean that $\operatorname{cov} (D_0, D_n) = E(D_0D_n) - (E(D_0))^2 = 0 (n = 1,2, \cdots)$. Schematically then, we study the mapping $\{T_n\} \overset{\{ S_n\}/1}{\longrightarrow} \{D_n\}$, and as consequences of the formal theorems of the paper the following statements can be justified $(T, S$, and $D$ denote typical members of $\{T_n\}, \{S_n\}$ and $\{D_n\})$. (i) $\operatorname{var} (D) \geqq \operatorname{var} (S)$, with equality only in the trivial case where both $\{T_n\}$ and $\{S_n\}$ are deterministic. (ii) Locally, the mapping can be made any of variance increasing, variance preserving, or variance decreasing (that is, all cases of $\operatorname{var} (D) >, =, < \operatorname{var} (T)$ are possible) by appropriate choice of $\{T_n\}$ and $\{S_n\}$. Globally however, the mapping is variance preserving, that is, $\operatorname{var} (D_1 + \cdots + D_n)/\operatorname{var} (T_1 + \cdots + T_n) \rightarrow 1\quad (n \rightarrow \infty)$. (iii) When $\{T_n\}$ is a Poisson process, the process $\{D_n\}$ is uncorrelated if and only if it is a Poisson process (and this occurs if and only if the $\{S_n\}$ are negative exponential). (iv) When the $\{S_n\}$ are negative exponential, $\{D_n\}$ is a renewal process if and only if it is a Poisson process (and this occurs if and only if $\{T_n\}$ is a Poisson process). However (cf. (iii)) $\{D_n\}$ can be uncorrelated without being a renewal process. If the $\{D_n\}$ are correlated then the terms $\operatorname{cov} (D_0, D_n)$ are of the same sign for all $n = 1,2, \cdots$ and converge to zero monotonically. (v) There exist $\{T_n\}$ and $\{S_n\}$ such that the serial covariances $\operatorname{cov} (D_0, D_n)$ are not of the same sign for all $n = 1,2, \cdots$ (see remark after Theorem 7).