Abstract
Most of the analysis done so far assumed a FCFS queueing regime. Yet some, indeed many, of our results (for example, those concerning mean waiting times) hold for any work-conserving, nonanticipating and nonpreemptive regime. For example, instead of a FCFS regime, we can assume a random regime such that, upon service completion, the next to commence service is selected at random from all those waiting. Another example is the last-come first-served without preemption regime (LCFS), where the next to commence service is the one who has been waiting the least. Indeed, in all of these entrance policies for the G/G/1 queue, the distribution of the number in the system, and hence, by Littleās law, the mean waiting time, is invariant with the queueing regime. Of course, many differences exist. For example, the variance of the waiting time in a FCFS queue is smaller than the corresponding value under LCFS. See [33].
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Notes
- 1.
Given that one is in stageĀ d, this is oneās last stage with probability p d āāāq d . The complementary event is that one moves to stageĀ dā+ā1 (once stageĀ d is completed) with probability q dā+ā1āāāq d .
- 2.
As always, it is only here that we define the model formally.
- 3.
SeeĀ [23] for two more properties for the case where Ļ(n) is a whole number (representing the case of a number of identical servers) and p(i,ān)ā=āĪ³(i,ān), with both parameters being equal to 1āāāĻ(n) or to zero, 1āā¤āiāā¤ān. This means that customers are assigned to dedicated servers.
- 4.
The permutation Ļ (n) permutes the index (1,ā2,āā¦,ān) into the index (Ļ (n)(1),āĻ (n)(2),āā¦,āĻ (n)(n)).
- 5.
We use Ļ (n) āāā1 as the standard notation for the inverse permutation of Ļ (n). In other words, Ļ (n) āāā1(Ļ (n)(1,ā2,āā¦,ān))ā=ā(1,ā2,āā¦,ān).ā
- 6.
This observation was communicated to us by Liron Ravner.
- 7.
This result appeared first inĀ [45]. We present here a different proof.
- 8.
Note that this product is invariant with the permutation.
- 9.
This means that a reshuffling of the indices takes place also at service completions.
References
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Haviv, M. (2013). Insensitivity and Product-Form Queueing Models. In: Queues. International Series in Operations Research & Management Science, vol 191. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6765-6_11
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