On preemptive-repeat LIFO queues

Abstract

In this paper, we study the basic properties of last-in first-out (LIFO) preemptive-repeat single-server queues in which the server needs to start service from scratch whenever a preempted customer reaches the server. In particular, we study the question of when such queues are stable (in the sense that the equilibrium time-in-system is finite-valued with probability one) and show how moments of the equilibrium customer sojourn time can be computed when the system is stable. A complete analysis of stability is provided in the setting of Poisson arrivals and in that of the Markovian arrival process. The stability region depends upon the detailed structure of the interarrival and service time distributions and cannot be expressed purely in terms of expected values. This is connected to the fact that such preemptive-repeat queues are not work conserving.

Appendix: Computational experience

The computations of Sect. 3.1 were done in MATLAB, using the iterative scheme (3.3) to compute $\varvec{P}$. In each step, $\varPsi (\varvec{P}_k)$ was evaluated via (3.1) using matrix inversion, rather than the power series (3.2). Matrix exponentials and the infinite integral were evaluated via MATLAB’s routines expm, resp. integral ($\cdot$,0,Inf,’ArrayValued’,true).

The deviation of $\varvec{P}$ from being stochastic was measured via two criteria, the deficit $\delta =$ $1-\varvec{e}^\top \varvec{P}\varvec{e}/q$ of the average row sum from 1 and calculated values of $\mathbb {E}D$. Ideally, one expects a change point at the critical point for stability, where $\log _{10}\delta$ should jump from $-\infty$ to finite negative values and $\mathbb {E}D$ should go to $\infty$.

An illustration of the procedure is given in Fig. 1 for the M/M/1 case where the critical value is known as $\rho =0.50$ (cf. Remark 2). Three numbers $K=100,\,500,\,5000$ of iterations were used. It shows that (1) a sufficiently large value of K is crucial for getting the desired sharp distinction between stability and non-stability. One also notes that (2) $\log _{10}\delta$ is not $-\infty$ in the numerics, but the wiggles for small $\rho$ indicates the numerical precision on $\delta$ is about $10^{-15}$, (3) The algorithm does not produce $\infty$ above the critical point, but some other number. This is no contradiction, since the expression $\varvec{m}\,=\,(\varvec{I}-\varvec{A}_{21})^{-1}\varvec{A}_{2a}$ is only valid assuming stability. The entries beyond that do not give $\mathbb {E}_i[D;\,D<\infty ]$ but rather the ratio of two integrals without any direct interpretation in terms of the queueing problem.

Fig. 1

M/M/1

The numbers given in the E$_p/$M/1 and H$_2/$M/1 examples were visually assessed from plot similar to Fig. 1. A general feature was that for $p\ge 2$ states the change at the critical value was somewhat less sharp than for $p=1$ (as in M/M/1) with the same number K of iterations, so that sometimes up to $K=20,000$ iterations were required around the critical value; note that the interpretation of $\varvec{P}_k$ in terms of the tree depth indicates that particularly many iterations are needed here. Obviously, the iteration scheme is time-consuming since each step involves computing an infinite integral of a function involving a matrix inverse and matrix exponentials. Finite machine precisions set further limit. Nevertheless, the results look reliable to us up to the given number of digits. One of the most demanding examples is presented in Fig. 2.

Fig. 2

H$_2$ arrivals, $\theta =1/8$, $\eta =14.6$