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Malyshev’s theory and JS-queues. Asymptotics of stationary probabilities. (English) Zbl 1039.60082
The paper considers a discrete-time nearest neighbor random walk (SHC random walk) in \(\mathbb{Z}^2_+\), whose jump probabilities are homogeneous in time and possess homogeneity relative to space shifts and symmetry relative to the reflection about the diagonal. These walks can describe the so-called join-the-shorter-queues (JS-queues). In JS-queues, tasks arrive within three independent Poisson processes \(\Xi_1\), \(\Xi_2\) and \(\Xi'\). Processes \(\Xi_1\) and \(\Xi_2\) are of rate \(\lambda\) and \(\Xi'\) of rate \(\lambda'\). Tasks from \(\Xi_1\) go to server 1, tasks from \(\Xi_2\) go to server 2 and tasks from \(\Xi'\) choose the shortest queue. The service rates at each server are equal 1, and tasks are served according to a conservative discipline (say, FCFS), without interruption. The paper gives a criterion for positive recurrence of SHC random walk and analyzes geometric asymptotics of the stationary probabilities \(\rho_{m+n, m}\) as \(m,n\to\infty\), \((m+ n)/n\sim \text{ctg\,}\gamma\).

MSC:
60K25 Queueing theory (aspects of probability theory)
60G50 Sums of independent random variables; random walks
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30F10 Compact Riemann surfaces and uniformization
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