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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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