Abstract
Analysis of Queueing Systems with an Infinite Number of Servers and a Small Parameter
Highlights
The current study of service networks with complex sending discipline in [1,2,3] transport networks [4,5,6] and the asymptotic behavior of Jackson networks [7] handled with the problematic of verifying the global convergence of the solutions of certain infinite systems of ordinary differential equations to a time-independent solution
We suppose that service time has mean 1/μ with exponential distribution
There is a stationary probability distribution for the states of the system and if N → ∞ the evolution of the values uk(t) becomes deterministic and the Markov chain asymptotically converges to a dynamic system the evolution of which is described by system of ordinary differential equations of infinite order uk(t) = μ (uk+1(t) − uk(t)) + λ (︀(uk−1(t))2 − (uk(t))2)︀
Summary
The current study of service networks with complex sending discipline in [1,2,3] transport networks [4,5,6] and the asymptotic behavior of Jackson networks [7] handled with the problematic of verifying the global convergence of the solutions of certain infinite systems of ordinary differential equations to a time-independent solution. We consider the dynamics of large-scale queueing systems that consist of infinite number of servers with a Poisson input flow of requests of intensity N λ. We suppose that service time has mean 1/μ with exponential distribution In this case a share uk(t) of the servers that have the queues lengths with not less than k can be described using a system of ordinary differential equations of infinite order. We investigate the truncation system of this ordinary differential equations of infinite order with a small real parameter order N. Tikhonov type Cauchy problem for this truncation system with small parameter ε and initial conditions is used for the simulation of behavior solutions and for analysis of large-scale queueing systems with taking into account parameters λ, μ, ε
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