Nikolaos Limnios - Queueing Theory 2

We consider a continuous-time queueing system with regenerative input flow X and m heterogeneous servers that may be not available for operation from time to time. We also propose that the velocity of the service may be dependent on the state of the server. Assume that for the i th server a stochastic process ni ( t ) with state space is defined. If ni ( t ) = 0, then the i th server is in unavailable state, for instance it is broken; if then the i th server is working with the velocity Service times of customers by the i th server in the case when the velocity of the service is equal to one constitute a sequence of iid random variables, which does not depend on the input flow and service times by other servers,

It is possible that an unavailable period starts while a customer is receiving service. Then service of the customer is immediately interrupted. There are various disciplines for continuation of the service after restoration (Gaver 1962). Here, we consider the preemptive resume service discipline assuming that interrupted service continues when the server returns from a blocked period and the service velocity is the next state of the process ni ( t ).

CONDITION 1.6.–The stochastic process is strongly regenerative with regeneration points with an exponential phase so that We also assume that

It follows from condition 1.6 and Smith’s (1955) theorem that there exist the limits

and

where ji takes values

To define an auxiliary process Yi ( t ) for the i th server, we introduce a counting process

Then

[1.10]

and

CONDITION 1.7.– Service times have the first exponential phase, i.e.

where and independent random variables and

As regeneration points for Y we take subsequence of the sequence such that at time interrupted services for processes are in the exponential phase. Because of conditions 1.6 and 1.7, Y is a strongly regenerative flow and we may define the common sequence of regeneration points for both processes X and Y with the help of formula [1.4]. We need only to take instead of

Because of [1.10]we can easily obtain from the renewal theory the formula for the rate of the auxiliary process

[1.11]

Now we may calculate the traffic rate ρ and under some assumptions we get the necessary and sufficient stability condition for the system based on theorems 1.1 and 1.2. As an example, we consider the famous case (Morozov et al . 2011) when i.e. a server may be in an available or unavailable state. Let be moments of breakdowns and moments of restorations for the i th server. Here