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Queueing Theory 2. Nikolaos Limnios

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Queueing Theory 2 - Nikolaos Limnios

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Now we construct the sequence
of common points of regeneration for processes X and Y with the help of [1.3]. Because of lemma 1.1
and the traffic rate ρ of the system is defined by [1.7].

      1 i)

      2 ii) Q(t) is a stochastically bounded process if ρ < 1.

      PROOF.– The first statement follows from theorem 1.1 since conditions 1.2 and 1.3 are realized.

      Let ρ < 1. For the system S, we introduce the embedded process

where Qn is the number of customers in the system on time Tn and ζi(n) = 1 if there is a customer on the ith server and ζi (n) = 0 otherwise. In a view of the service discipline after service restoration and properties of the synchronization epochs
the process
is a Markov chain with countable set of states
j > m}. Let R0 be the set of unessential states and
irreducible classes of communicating states. It follows from the condition ρ < 1 that the number of classes r < ∞.

      For any aperiodic class

of states based on Foster’s criterion (Meyn and Tweedie 2009), we may easily prove that this class is ergodic (Afanasyeva and Tkachenko 2016, 2018). Therefore, the process Qn is stochastically bounded if
It is also true if
is a periodic class. Since the number of classes r < ∞, we obtain the stochastic boundedness of the process Qn and therefore Q(t).

      We may obtain the upper bound of the traffic rate ρ providing the stochastic boundedness of the process Q. It is known from (Borovkov 1976) that

      Therefore

      and sufficient condition of the stochastic boundedness of Q has the following form

      If bi = b, then we have the same condition as obtained in Morozov et al. (2011) ■

      In this section we study a continuous-time queueing system with two independent regenerative input flows X1 and X2 with intensities λ1 and λ2 and m servers. The customers of the second type (which belong to X2 ) have an absolute priority with respect to customers of the first type. Service interruption for the low priority customer occurs when a high priority customer arrives during a low priority customer’s service time. If at an arrival time of the second type customer there are m1 free servers, m2 servers occupied by customers of the first type and m – m1 – m2 servers occupied by customers of the second type, then an arriving customer randomly chooses any server from m1 + m2 servers, which are not busy by customers of the second type. Service times by the ith server for high(low) priority customers have distribution function B0

with mean
Therefore, for high priority customers we have a system Reg|G|m with homogeneous servers and for low priority customers a system with interruptions and preemptive resume service discipline considered in section 1.6.

      Denote by Qi(t) the number of customers of the ith type at the system including the customers on the servers at time

and
be the sequences of regeneration points for X1 and X2, respectively. Under some additional conditions, for example, when the inequality [1.14] is valid for the function B0 (other sufficient assumptions are given in Afanasyeva and Tkachenko (2014)), the process Q2 is regenerative with points of regeneration

      The stability condition for the process Q2 has the form (Afanasyeva and Tkachenko 2014)

      that is supposed to be fulfilled. We now want to get the stability condition for the process Q1.

As regeneration points for
, we take subsequence
of the regeneration points sequence
for the input flow X2 such that
As before, we assume that [1.14] holds for B0. Since X2 is a strongly regenerative flow, condition 1.6 is fulfilled.

      To obtain the traffic rate for low priority customers, we need to find

Because of the rule of the server choose by an arriving high priority customer, we have
for all
To calculate π, we define for high priority customers the following processes. Let wi(t)

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