Nikolaos Limnios - Queueing Theory 2

[1.12]

Then denote the length of the n th blocked and the n th available period for the i th server, respectively, The sequence consists of iid random vectors (for all ) and these sequences do not depend on the input flow X and service times. Let be the length of the n th cycle for the server i . A cycle consists of a blocked period followed by an available period. We assume that

We put ni ( t ) = 0 if the i th server is in an unavailable state at time t and ni(t) = 1, otherwise If a blocked period has an exponential phase, i.e. where are independent random variables and has an exponential distribution with a parameter αi , then we may define the sequence of regeneration points for the regenerative process as above. Therefore, condition 1.6 holds. Under condition 1.7, the auxiliary process Y is strongly regenerative and we can construct the common points of regeneration for X and Y and apply theorems 1.1 and 1.2 for this model. Since

we have from [1.11]

If bi = b , then we get the same stability condition as obtained in Morozov et al . (2011) for a queueing system GI|G|m with a common distribution function of service times for all servers.

COROLLARY 1.1.– For a queueing system with

if ρ > I.

Under condition 1.4, the process is stochastically bounded if ρ < 1.

PROOF.– Let, as before, be the number of customers actually served on the i th server up to time t . It is evident that stochastic inequality

for t > 0 takes place and hence

Since

To prove the second statement, we first assume that conditions 1.6 and 1.7 hold. Then condition 1.1 for the process Y takes place. We also may organize the performance of the systems S and S 0in such a way that inequality [1.8]is realized when Thus, conditions 1.1, 1.4 and 1.5 are satisfied and because of theorem 1.2 the process Q is stochastically bounded.

If conditions 1.6 and 1.7 (or one of them) are not valid, we construct a system Sδ satisfying conditions 1.6 and 1.7 and majorising our system S , so that in distribution

[1.13]

Here, Qδ ( t ) is the number of customers in the system Sδ at instant t . Let us introduce independent sequences of iid random variables with exponential distribution with a rate δ . Assume that repair time in the system Sδ has the form and service time by the i th server has the form

Then Sδ satisfies conditions 1.6 and 1.7. Since and we may choose δ so that ρδ < 1.