On the rate of convergence for infinite server Erlang–Sevastyanov’s problem

Abstract

Polynomial convergence rates in total variation are established in Erlang-Sevastyanov's type problem with an infinite number of servers and a general distribution of service under assumptions on the intensity of serving.

About quasi-renewal processes and quasi-regenerative processes

“…In the first attempts to analyse the recurrence property of the piecewise linear processes in queuing theory, the special Lyapunov function was used. Thus, all distributions of all renewal times defining the queuing process had to be absolutely continuous ( [9,10]). But the use of Lorden's inequality made it possible to find an upper bound for the rate of convergence of the distribution of the queuing process ( [21]).…”

“…Probably, the first case of the analysis of such a system was given in [9]. Now, the analysis of such systems is given in [10][11][12][13][14][15]19] and others. is named quasi-renewal process if the r.v.…”

“…In [9][10][11][13][14][15], the coupling method was used for the analysis of the recurrence properties of some queuing and reliability systems. But in this paper, there is no algorithm for the calculation of an upper bound for the constant C(ℓ).…”

On Strong Bounds of Rate of Convergence for Regenerative Processes

“…Our goal is to find the bounds of the constants C (X 0 , k) and C (X 0 , a) in sufficiently wide conditions; note that the behavior of the constants C (X 0 , k) and C (X 0 , a) has been studied in [10,11,15,16,17,18] for some special cases of regenerative processes.…”

On convergence rate for Erlang--Sevastyanov type models with infinitely many servers

“…The paper [21] was based on the MSc project by the author on the topic suggested by Professor A.D. Solovyev. The ultimate goal of that time -to estimate rates of convergence for Erlang-Sevastyanov systems -was partially realised only recently [23] where a polynomial convergence for such systems with finitely or infinitely many servers has been established under certain assumptions on the intensities; a bit earlier in [12] exponential rate was proved under more severe than in [23] assumptions. The problem of evaluating convergence rates without assuming anything about intensities apparently remains open.…”