A Large Deviation Principle for Join the Shortest Queue
Mathematics of Operations Research
Markov Chains and Stochastic Stability
Markov Chains and Stochastic Stability
Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks
Mathematics of Operations Research
Asymptotic behavior for MAP/PH/c queue with shortest queue discipline and jockeying
Operations Research Letters
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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 two 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.