Queueing Theory 2. Nikolaos Limnios

Queueing Theory 2 - Nikolaos Limnios


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Q(t) is a stochastically bounded process, i.e.

      2 ii)

      CONDITION 1.5.– If

, then for any ϵ > 0 there exists such that for n > nϵ

      THEOREM 1.1.– Let conditions 1.3 and 1.1 (1.2) for the continuous-time (for the discrete-time) case be fulfilled. If ρ ≥ 1, then

      [1.9]

      Define the auxiliary sequence

recursion

      Because of the equality

and condition 1.3 we get the stochastic inequality
It is well known (Feller 1971) that

      and in distribution

      For ρ ≥ 1, the sequence

is a random walk with a non-negative drift. Hence, except when
(c is a constant),
(w.p.1) (Feller 1971) that completes the proof. ■

      Our objective here is to establish stochastic boundedness of the process Q when the traffic rate ρ < 1. Under some additional assumptions providing the regenerative structure of the process Q, this property has its stability as a consequence.

      THEOREM 1.2.– Let conditions 1.4and 1.5 and condition 1.1 (1.2) for the continuoustime (for the discrete-time) case be fulfilled. If ρ < 1, then Q is a stochastically bounded process.

      PROOF.– Because of condition 1.4 there are two possible cases: Qn = Q(Tn) is either stochastically bounded or

Assume that the second case takes place and ρ < 1. Because of condition 1.5 for
there is such that for n > nϵ

      that contradicts our assumption that

      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 ith server a stochastic process ni(t) with state space

is defined. If ni(t) = 0, then the ith server is in unavailable state, for instance it is broken; if
then the ith server is working with the velocity
Service times of customers by the ith 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

      where ji takes values

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

      Then


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