Instability of LIFO queueing networks

Abstract

Under the last-in, first-out (LIFO) discipline, jobs arriving later at a class always receive priority of service over earlier arrivals at any class belonging to the same station. Subcritical LIFO queueing networks with Poisson external arrivals are known to be stable, but an open problem has been whether this is also the case when external arrivals are given by renewal processes. Here, we show that this weaker assumption is not sufficient for stability by constructing a family of examples where the number of jobs in the network increases to infinity over time. This behavior contrasts with that for the other classical disciplines: processor sharing (PS), infinite server (IS) and first-in, first-out (FIFO), which are stable under general conditions on the renewals of external arrivals. Together with LIFO, PS and IS constitute the classical symmetric disciplines; with the inclusion of FIFO, these disciplines constitute the classical homogeneous disciplines. Our examples show that a general theory for stability of either family is doubtful.

Appendix

As mentioned in Subsection 1.1, the processor sharing (PS) discipline is stable for all subcritical queueing networks with Poisson input and exponentially distributed service times; its equilibrium distribution can be written explicitly in its famous “product form”. As with symmetric queueing networks in general, the exponentially distributed service law can be relaxed to a mixture of Erlang distributions. (An Erlang distribution is the distribution of a sum of i.i.d. exponentially distributed random variables.) This generalization employs the method of stages (see, for example [6, 9]), and the resulting equilibria are again explicit. It is not difficult to check that mixtures of Erlang distributions are dense in the weak topology in the set of distribution functions (see, for example Exercise 3.3.3 in [9]).

Consider a subcritical queueing network with Poisson input, but arbitrary service distribution. Because of the explicit nature of the equilibria for mixtures of Erlang distributions, one can choose a sequence of queueing networks whose service distributions converge to the service distributions of the given network and the corresponding sequence of equilibria is tight. Because of this, the above representation of equilibria extends to all subcritical PS queueing networks with Poisson input [1]. In particular, these queueing networks with general service distributions are stable.

This explicit representation no longer holds when external arrivals are generalized to renewal processes. However, for networks with general interarrival distributions (assuming only (59) and (60) below) but where the service times are exponentially distributed, one can compare PS networks with networks with the head-of-the-line proportional processor sharing (HLPPS) discipline. The service rule for the HLPPS discipline is the same as that for PS, except that all service a class receives is devoted to the earliest arriving job at that class, rather than being spread out uniformly among jobs at that class. When the service distributions are exponentially distributed, the specific rule assigning service within a class does not affect the rate at which jobs leave the class, and so processes with the PS and HLPPS disciplines and exponentially distributed service times have the same law.

The stability of subcritical HLPPS networks can be shown under interarrival distributions satisfying (59) and (60), and general service distributions by employing the standard machinery of fluid limits (Corollary 1 of Theorem 1, in [5]). This technique unfortunately produces little qualitative information about the nature of the equilibria for the corresponding queueing networks.

In this appendix, we make two observations. First, that the connection between the PS and HLPPS disciplines, together with the method of stages, enables one to quickly show, using known results, the stability of subcritical PS networks for interarrival times satisfying (59) and (60) and a dense family of service distributions. This is done in Proposition 9 below.

The second observation is that it nevertheless appears to be difficult to extend this result to all service distributions. One cannot use the above argument that was applied for Poisson input without somehow first showing the tightness of the equilibria for the corresponding sequence of queueing networks. Because of the absence of a direct characterization of these equilibria, it is not clear how to proceed. Nevertheless, based on “obvious” intuition, such stability should hold for subcritical PS queueing networks with both general renewal input and service distributions.

For Proposition 9, we first state Corollary 1 of Theorem 1 [5] when the service times are exponentially distributed. Two assumptions on the interarrival times are required: The interarrival time distributions are unbounded, that is, denoting by $\xi _k (i)$, $i\in {\mathbb {Z}}_+$, the i.i.d. interarrival times at a class k with external arrivals,

$$\begin{aligned} P(\xi _k(1) \ge x) > 0 \quad \text { for all } x. \end{aligned}$$

(59)

Moreover, for some $\ell _k > 0$ and non-negative $q_k(\cdot )$, with $\int _0^{\infty }q_k(x)dx > 0$,

$$\begin{aligned} P(\xi _k(1) + \cdots +\xi (\ell _k) \in dx) \ge q_k(x) dx, \end{aligned}$$

(60)

that is, the above sum dominates Lebesque measure in an appropriate sense. We note that both properties are only needed to ensure that all states communicate with one another; they are not needed to show that the total number of jobs in the network Z(t) is tight (without these conditions, the residual interarrival times could synchronize in some manner).

Proposition 8

Any subcritical HLPPS queueing network with exponentially distributed service times and whose external interarrival times satisfy (59) and (60) is stable.

In Sect. 1, LIFO queueing networks, with routing given by Fig. 1, were reinterpreted as LIFO networks, with routing given by Fig. 2, by decomposing the service time at a class into service times at successive classes at the same station. Since the PS discipline is symmetric, the same reasoning can be applied to it as well; in fact, since service is assigned uniformly to all jobs within a station, it is easy to see that any reclassification of classes within a station will not affect service.

This reasoning can be applied to service times that are mixtures of Erlang distributions, as in the method of stages. One thus extends results on the equilibria for subcritical PS networks with renewal external arrivals and exponentially distributed service times to subcritical PS networks with renewal external arrivals and service times that are mixtures of Erlang distributions. This reasoning was applied for Poisson external arrivals (see, for example [9]). In the current setting, one does not obtain an explicit formula for equilibria, but stability nevertheless follows:

Proposition 9

Any subcritical PS queueing network whose external interarrival times satisfy (59) and (60) and whose service times are mixtures of Erlang distributions is stable.

Proof

By Proposition 8, any HLPPS network whose external interarrivals satisfy (59) and (60) and whose service times are exponentially distributed is stable. By the above reasoning, the same queueing network, but with the PS rather than the HLPPS discipline, is also stable. On account of the PS discipline, the evolution of a queueing network will be the same when classes at a given same station are combined into a single class. It follows from this that a subcritical PS network whose external interarrival times satisfy (59) and (60) and whose service times are mixtures of Erlang distributions will be stable. $\square$