An Exact Analysis of the Multi‐classM/M/kPriority Queue with Partial Blocking

Abstract

We consider a multi-server queuing model with two priority classes that consist of multiple customer types. The customers belonging to one priority class customers are lost if they cannot be served immediately upon arrival. Each customer type has its own Poisson arrival and exponential service rate. We derive an exact method to calculate the steady state probabilities for both preemptive and nonpreemptive priority disciplines. Based on these probabilities, we can derive exact expressions for a wide range of relevant performance characteristics for each customer type, such as the moments of the number of customers in the queue and in the system, the expected postponement time and the blocking probability. We illustrate our method with some numerical examples.

Queuing with Heterogeneous Users: Block Probability and Sojourn times

“…To the best of our knowledge we are not aware of a work that directly studies this type of a heterogeneous achievable region. Some variants of queueing systems (e.g., [3], [6], [8], [12], [11], [13], [17] etc) have some connections to few parts of our models, and these are discussed in [15].…”

“…In [12] authors consider multi-class queueing system with eager and tolerant customers. This is the queueing system that is closest to the one considered in this paper, especially to CD policy.…”

“…With our CD policy, the tolerant customers utilize all the remaining servers, and hence the system is work conserving (only) with regard to the tolerant customers. While in [12] tolerant customers are also served in multi-server mode, i.e., each tolerant customer is provided one server and the idle servers (if any) are not utilized. Further the authors in [12] obtain a set of (balance) equations, solving which stationary probabilities (and then the stationary performance) can be derived.…”

“…While in [12] tolerant customers are also served in multi-server mode, i.e., each tolerant customer is provided one server and the idle servers (if any) are not utilized. Further the authors in [12] obtain a set of (balance) equations, solving which stationary probabilities (and then the stationary performance) can be derived. We provide a closed form expression for these performance measures in fluid limit for eager customers.…”

“…However the challenging part is to obtain the performance for the tolerant customers. One can obtain system of equations (as in [12]), the solution of which provides the performance measure. However once the pseudo conservation law is proved, the performance of tolerant customers is immediately known as soon as the performance of the eager customers is computed.…”