Читать книгу Queueing Theory 2 - Nikolaos Limnios - Страница 14

For the system S, we define an auxiliary system S₀ with input flow X₀ such that when the number of customers in the system becomes less than m a new customer immediately arrives in the system. Therefore, there are always customers for service in S₀. Other characteristics such as the initial state, the sequence stochastic process and a functional Φ are the same as for the system S. If in the system S the initial number of customers Q (0) < m, then the process X₀ has the jump m – Q(0) at zero instant. We determine an auxiliary service process Y(t) as the number of customers served in S₀ during (0, t). Since the flow Y is defined by the processes and V and these processes do not depend on the input flow X at the system S, we conclude that X and Y are independent flows.

We also need additional assumptions.

CONDITION 1.1.– For the continuous-time case, Y is a strongly regenerative flow with the sequence as points of regeneration.

We call the regenerative flow Y strongly regenerative if the regeneration period has the form

[1.2]

where are independent random variables and

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 for both processes X and Y letting in the discrete-time case

[1.3]

and in the continuous-time case

[1.4]

LEMMA 1.1.– Let for the continuous-time (discrete-time) condition 1.1 (condition 1.2) be fulfilled. Then the sequence consists of common regeneration points for X and Y and

[1.5]

for the continuous-time case,

[1.6]

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 Then is a sequence of iid random variables and in accordance with Wald’s identity (Feller 1971). Therefore, we need to prove the finiteness of Eν1. Denote by h₂(t) (h(t)) the mean number of renewals at time t for the renewal process so that

and

Taking into account condition 1.2, we derive from Blackwell’s theorem (Thorisson 2000)

Because of X and Y independence

Since w.p.1, then w.p.1. Therefore, from Lebesgue s dominated convergence theorem, we obtain ■

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:

[1.7]

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 as the number of customers served at the system S during time interval [0, t).

CONDITION 1.3.– The following stochastic inequalities take place:

Let Q(t) be the number of customers in the system S including the customers on the servers at time t so that

CONDITION 1.4.– There are two possible cases:

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

2 ii)

CONDITION 1.5.– If , then for any ϵ > 0 there exists nϵ such that for n > n_ϵ

[1.8]