arXiv subjects

Masahiro Kobayashi

Publications and source records attributed to Masahiro Kobayashi.

21 records · Page 2Linked to original sources

arXiv2014

Structure-reversibility of a two dimensional reflecting random walk and its application to queueing network

We consider a two dimensional reflecting random walk on the nonnegative integer quadrant. It is assumed that this reflecting random walk has skip free transitions. We are concerned with its time reversed process assuming that the stationary distribution exists. In general, the time reversed process may not be a reflecting random walk. In this paper, we derive necessary and sufficient conditions for the time reversed process also to be a reflecting random walk. These conditions are different from but closely related to the product form of the stationary distribution.

math.PR↗

arXiv2013

Join the shortest queue among $k$ parallel queues: tail asymptotics of its stationary distribution

We are concerned with an $M/M$-type join the shortest queue ($M/M$-JSQ for short) with $k$ parallel queues for an arbitrary positive integer $k$, where the servers may be heterogeneous. We are interested in the tail asymptotic of the stationary distribution of this queueing model, provided the system is stable. We prove that this asymptotic for the minimum queue length is exactly geometric, and its decay rate is the $k$-th power of the traffic intensity of the corresponding $k$ server queues with a single waiting line. For this, we use two formulations, a quasi-birth-and-death (QBD for short) process and a reflecting random walk on the boundary of the $k+1$-dimensional orthant. The QBD process is typically used in the literature for studying the JSQ with 2 parallel queues, but the random walk also plays a key roll in our arguments, which enables us to use the existing results on tail asymptotics for the QBD process.

math.PR↗

arXiv2012

Revisit to the tail asymptotics of the double QBD process: Refinement and complete solutions for the coordinate and diagonal directions

We consider a two dimensional skip-free reflecting random walk on a nonnegative integer quadrant. We are interested in the tail asymptotics of its stationary distribution, provided its existence is assumed. We derive exact tail asymptotics for the stationary probabilities on the coordinate axis. This refines the asymptotic results in the literature, and completely solves the tail asymptotic problem on the stationary marginal distributions in the coordinate and diagonal directions. For this, we use the so-called analytic function method in such a way that either generating functions or moment generating functions are suitably chosen. The results are exemplified by a two node network with simultaneous arrivals.

math.PR↗