the residual service time (virtual waiting time) on the ith server at instant t and Zi(t) the total service time of customers which arrived up to time t and have to be served on the ith server. Thus
where ηj is the service time of the jth arrived customer. We note that w.p.1
Therefore, the traffic rate for low priority customers has the form
From corollary 1.1 we obtain corollary 1.3.
COROLLARY 1.3.– Assume that X2 is a strongly regenerative flow, B0 satisfies [1.14] and
Then
If additionally Bi satisfies [1.14] and ρ1 < 1, then Q1(t) is a stable process.
The first statement follows from corollary 1.1 since
is a strongly regenerative process. To prove the second statement, we note that Q2 is a regenerative process and its stochastic boundedness means stability.
1.9. Queueing system with simultaneous service of a customer by a random number of servers
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
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
the residual service time (virtual waiting time) on the ith server at instant t and Zi(t) the total service time of customers which arrived up to time t and have to be served on the ith server. Thus
