Читать книгу Queueing Theory 2 - Nikolaos Limnios - Страница 20

Here, we consider a system S assuming that the nth customer requires service from ζn server simultaneously The customer arrived in an empty queue begins service immediately if the number of available servers is more or equal ζn. Otherwise the customer becomes the first in a queue and the service begins when the required number of servers becomes available. A customer who arrives in an nonempty queue takes the last place in the queue. When service begins, each server’s completion time is independent of all other servers and has an exponential distribution with rate μ. The sequence consists of iid random variables and .

Queueing systems with simultaneous service have been studied in a number of works (Rumyantsev and Morozov 2017; and references therein). The stability conditions in an explicit form have been obtained for systems with a Poisson input flow and independent exponentially distributed service times by (Gillent and Latouche 1983). The main goal of this section is an extension of the stability condition to the model with a regenerative input flow X based on theorems 1.1 and 1.2. Thus, we consider the system S described in section 1.2 with is a sequence of independent exponentially distributed random variables not depending on ζn. Let S₀ be an auxiliary system defined in section 1.3. Instead of the process Y we consider the auxiliary flow Z that is the number of service completions by all m servers up to time t in the auxiliary system S₀. Denote by U(t) the number of occupied servers at time t in the system S₀. Then U is a Markov chain and Z is a doubly stochastic Poisson process (Grandell 1976) with a random intensity We note that the process U hits the state {m} from any state j = 1, 2,..., m with a positive probability. It means that all states attainable from the state {m} constitute the finite class K of communicating states. Therefore, there are limits

Let and

We have the system of equations for

[1.18]

where We may easily verify that the solution of [1.18] has the form

Since

we get

and the traffic rate for the system S

[1.19]

Let us note that this formula is the same as obtained by (Gillent and Latouche 1983) for queueing systems with a Poisson input flow. To employ theorems 1.1 and 1.2, we have to verify conditions 1.1, 1.3, 1.4 and 1.5. First of all we note that Z is a strongly regenerative flow. As points of regeneration, we may take the sequential hitting times into the fixed state (We take if Then is the sojourn time in the state j and is the return time to this state after exit from it for U(t). The random variables and are independent and has an exponential distribution. Moreover, therefore, condition 1.1 holds. Let q(t) be the total number of servers that are already busy or will be busy by service of the Q(t) customers, which are present at the system S at time t. Then q(t) as well as Q(t) is a regenerative process with a sequence of points of regeneration that is a subsequence of such that Let us recall that is a sequence of points of regeneration for X. Therefore, because of theorem 1 from (Afanasyeva and Tkachenko 2014), condition 1.4 is fulfilled. Now for any fix we define the common points of regeneration for the input flow X and auxiliary flow Z by the relation

Let

[1.20]

where is the process Z for the system S that is the number of service completions by all m servers up to time t. Now we formulate the main result of this section.

COROLLARY 1.4.– For the system S, the process Q is a stable process if and only if ρ < 1.

PROOF.– Consider the case ρ ≥ 1 and take m instead of j in [1.20]. It is evident that the stochastic inequality

takes place. Therefore condition 1.3 is fulfilled. Based on theorem 1.1, we obtain the convergence

[1.21]

For the case ρ < 1, we take j0 instead of j in [1.20]. Then if convergence [1.21] takes place for any ϵ > 0, there is n_ϵ such that

The proof is based on the approach described in Afanasyeva and Tkachenko (2014). Thus, conditions 1.1, 1.4 and 1.5 of theorem 1.2 are fulfilled and Q is a stochastically bounded process when ρ < 1. Since Q is a regenerative process, this means that it is stable.

We see that for the model under consideration, the stability condition does not depend on the structure of the input flow. ■