Queueing Theory 2. Nikolaos Limnios

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) ■

1.8. Queueing system with a preemptive priority discipline

      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)

[1.17]

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

We start with the definition of the process of interruptions. Let ni(t) =0 if at instant t the ith server is occupied by a high priority customer and ni(t) = 1 otherwise,

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) be the residual service time (virtual waiting time) on the ith server at instant t and Zi(t) the total service time of customers which arrived up to time t and have to be served on the ith server. Thus

      where ηj is the service time of the jth arrived customer. We note that w.p.1

      and

because of the stability condition [1.17].

      Since

      and w.p.1

      then

Therefore, the traffic rate for low priority customers has the form

From corollary 1.1 we obtain corollary