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I'm not sure if this question is suited to MO. I will happily delete if not.

Situation

Consider a general queueing system $\mathscr{S}$, whose customer arrival times are independent, and whose service times are independent; both of these are allowed to have general distributions. No assumptions are made about whether not customers are served singly or in batches (or about the size of the batches in that case). The queue is also allowed, but not required, to have finite capacity.

The queue is explicitly assumed be a regenerative process.

Suppose that in certain situations as the queue evolves, the behavior of the service times or customer arrivals is allowed to be changed.
For example, the modified queue might occasionally be allowed to turn away a customer (whereas the original system $\mathscr{S}$ would typically accept them). Or perhaps the modified queue in certain situations might be able to eliminate the remaining service time for a server which is currently busy, so as to immediately complete the service for a customer (whereas these customers in the original system $\mathscr{S}$ would still be left being served).

EDIT: Example

Let $\mathscr{U}$ be a finite capacity $M/G/K/K$ queue, and let $\mathscr{T}$ be a finite capacity $M/G'/K'/K'$ queue, which both see the same identical customer demand realization (you can imagine that customers are Poisson arrivals of pairs of customers, with one routed to each queue). The behavior of queue $\mathscr{T}$ is changed so that all customers are turned away whenever the queue $\mathscr{U}$ is at full capacity.

Question

If there are only finitely many epochs when these policy changes occur, then the regenerative nature of the queue guarantees there is no effect on the long run behavior of the queue.

But if on the other hand there is a policy (say for example driven either by the state of the queue itself, or maybe the state of some other process) which allows these "one off" behavior changes to become recurring, then there likely will be a change to the long run dynamics.

In particular, if the policy changes have the short term effect of decreasing service time for customers, or of reducing the number of customer arrivals, can one say anything about the effect on long run average queue length?

Deep thanks for any information.

(In my situation, the queue in question is the outstanding replenishment orders for a lost sales inventory system with constant lead time.)

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    $\begingroup$ The paper by Sonderman "Comparing Multi-Server Queues with Finite Waiting Rooms, II: Different Numbers of Servers", Adv. in Applied Probability, 439--447, 1979, [JSTOR] (jstor.org/stable/1426848?seq=1) could help. $\endgroup$
    – user137846
    Sep 4, 2014 at 10:02
  • $\begingroup$ @user137846 : That was really quite helpful, at least as a starting point. That article points to a promising collection of articles related to stochastic comparisons of queues & semi-Markov processes. Thanks a bunch! $\endgroup$
    – bryanj
    Sep 5, 2014 at 1:58

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