Queues with path-dependent arrival processes

Abstract

We study the transient and limiting behavior of a queue with a Pólya arrival process. The Pólya process is interesting because it exhibits path-dependent behavior, e.g. it satisfies a non-ergodic law of large numbers: the average number of arrivals over time [0, t] converges almost surely to a nondegenerate limit as $t \rightarrow \infty$. We establish a heavy-traffic diffusion limit for the $\sum_{i=1}^{n} P_i/GI/1$ queue, with arrivals occurring exogenously according to the superposition of n independent and identically distributed Pólya point processes. That limit yields a tractable approximation for the transient queue-length distribution, because the limiting net input process is a Gaussian Markov process with stationary increments. We also provide insight into the long-run performance of queues with path-dependent arrival processes. We show how Little’s law can be stated in this context, and we provide conditions under which there is stability for a queue with a Pólya arrival process.

Heavy traffic limits for queues with non-stationary path-dependent arrival processes

“…This paper extends our paper [ 9 ], which established results for the special case of a stationary GPP. Theorem 1 of [ 9 ] shows that a GPP is a stationary point process (has stationary increments) if …”

“…Our goal in this paper was to see how much of this appealing tractability we could achieve without requiring the stationarity. We succeed in generalizing both the HTLT and a result obtained in Corollary 6 of [ 9 ] describing the transient distribution of the queue’s limit process. We also show that a general GPP can be approximated arbitrarily well by a piecewise-stationary GPP, and we use that structure to obtain an explicit transient approximation formula particularly convenient for modeling.…”

“…As discussed in [ 9 ], a stationary GPP satisfies a non-ergodic law of large numbers, which causes the queue length process (not normalized by time) to approach infinity with positive probability as time evolves. (For the stationary GPP, we defined path dependence by lack of ergodicity; for the more general GPP, we define path-dependence by the lack of ALOM.)…”

“…As discussed in [ 14 ], self-exciting point processes are interesting to capture overdispersion found in arrival process data, for example [ 12 ]. However, as discussed in Remark 3 of [ 9 ], the GPP is quite different from the Hawkes process; for example, unlike the GPP, the Hawkes process is always stationary and ergodic and has the ALOM property. The GPP can capture the long-term impact of initial conditions.…”

“…In our opinion, the greatest appeal of the heavy-traffic limit for the P/GI/1 queue with a stationary GPP arrival process in [ 9 ] is its remarkable tractability, as illustrated by the explicit three-dimensional distribution of the limit process in Corollary 6 of [ 9 ]. That tractability suggests that the GPP can be useful for modelling in practical applications.…”