Queueing Theory 2. Nikolaos Limnios
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We also need additional assumptions.
CONDITION 1.1.– For the continuous-time case, Y is a strongly regenerative flow with the sequence
We call the regenerative flow Y strongly regenerative if the regeneration period
[1.2]
where
CONDITION 1.2.– For the discrete-time case, processes X and Y are regenerative aperiodic flows. As usually, aperiodicity means that the greatest common divisor (GCD)
Then we may determine common points of regeneration
and in the continuous-time case
LEMMA 1.1.– Let for the continuous-time (discrete-time) condition 1.1 (condition 1.2) be fulfilled. Then the sequence
for the continuous-time case,
for the discrete-time case.
PROOF.– Since the proof of [1.5] is almost the same as the proof of [1.6], we consider the discrete-time case only. Let
so that
and
Taking into account condition 1.2, we derive from Blackwell’s theorem (Thorisson 2000)
Because of X and Y independence
Since
Later we consider both cases (discrete-time and continuous-time) together. We only have to take condition 1.2 instead of condition 1.1.
Let
Then
We define the traffic rate as follows:
We think of λX and λγ as the arrival and service rate, respectively. Intuitively, it is clear that the number of customers in the system S is a stochastically bounded process if ρ < 1 and it is not the case if ρ ≥ 1. The main stability result of this chapter consists of the formal proof of this fact.
We define the stochastic flow
CONDITION 1.3.– The following stochastic inequalities take place: