Abstract
We consider an asymmetric inclusion process, which can also be viewed as a model of n queues in series. Each queue has a gate behind it, which can be seen as a server. When a gate opens, all customers in the corresponding queue instantaneously move to the next queue and form a cluster with the customers there. When the nth gate opens, all customers in the nth site leave the system. For the case where the gate openings are determined by a Markov renewal process, and for a quite general arrival process of customers at the various queues during intervals between successive gate openings, we obtain the following results: (i) steady-state distribution of the total number of customers in the first k queues, $$k=1,\dots ,n$$k=1,ź,n; (ii) steady-state joint queue length distributions for the two-queue case. In addition to the case that the numbers of arrivals in successive gate opening intervals are independent, we also obtain explicit results for a two-queue model with renewal arrivals.
Highlights
The asymmetric inclusion process (ASIP), introduced and analyzed in [5,6,7,8,9], is a one-dimensional lattice of n sites, where particles arrive randomly into the first site (Q1), stay there (‘served’) for a random time, continue moving simultaneously and unidirectionally from site to site while staying for a random time in each site, until exiting the last site (Qn) and leaving the system
The ASIP defines the missing link between the celebrated Tandem Jackson Network (TJN) and the Asymmetric Exclusion Process (ASEP) [1,2,3] which plays the role of a paradigm in nonequilibrium statistical mechanics
Denoting by Ccapacity the capacity of a site, and by Cgate the capacity of the site’s gate, for the TJN, Ccapacity = ∞ and Cgate = 1, while for the ASEP Ccapacity = Cgate = 1
Summary
The asymmetric inclusion process (ASIP), introduced and analyzed in [5,6,7,8,9], is a one-dimensional lattice of n sites (queues), where particles (for example, customers) arrive randomly into the first site (Q1), stay there (‘served’) for a random time, continue moving simultaneously and unidirectionally from site to site while staying for a random time in each site, until exiting the last site (Qn) and leaving the system. The main contribution of this paper is that it considerably extends the exact analysis of ASIP tandem models: We allow the gate openings to be determined by a Markov renewal process, instead of assuming that each gate opens after exponentially distributed intervals, and we extend the Poisson arrival assumption by allowing a quite general arrival process of customers at the various queues during intervals between successive gate openings. Under these assumptions, we determine the steady-state distribution of the total number of customers in the first k queues, k = 1, .
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