Stochastic Complement Analysis of Multi - Server Threshold Queueswith

We consider a K-server threshold-based queueing system with hysteresis, for which a set of forward thresholds (F 1 ; F 2 ; : : : ; F K1) and a set of reverse thresholds (R 1 ; R 2 ; : : : ; R K1) are deened. A simple version of this multi-server queueing system behaves as follows. When a customer arrives to an empty system, it is serviced by a single server. Whenever the number of customers exceeds a forward threshold F i , a server is added to the system and server activation is instantaneous. Whenever the number of customer falls below a reverse threshold R i , a server is removed from the system. We consider and solve several variation of this problem, namely: (1) homogeneous servers with Poisson arrivals, (2) homogeneous servers with bulk (Poisson) arrivals, and (3) heterogeneous servers with Poisson arrivals. We place no restrictions on the number of servers or the bulk sizes or the size of the waiting room. In 8], the authors solve a limited form of this problem using the Green's function method. More speciically, they give a closed-form solution for a K-server system, when the servers are homogeneous, and for a 2-server system, when the servers are heterogeneous; the authors experienced diiculties in extending the Green's function method beyond the case of 2 heterogeneous servers. Rather than using Green's function, we solve this problem using the concept of stochastic complementation, which is a more intuitive and more easily extensible method. For the case of a homogeneous multi-server system we are able to derive a closed-form solution for the steady state probability vector; for the remaining cases we give an algorithmic solution. Note, however, that we can use stochastic complementation to derive closed-form solutions for some limited forms of cases (2) and (3), such as heterogeneous servers with K = 2 and bulk arrivals with a limited bulk size. Finally, our technique works both for systems with nite and innnite waiting rooms.