Queueing Theory 2. Nikolaos Limnios

      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 ith server. Here

[1.12]

      Then

denote the length of the nth blocked and the nth available period for the ith 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 nth 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 ith 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 ith 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 S₀ in 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]