Transform Methods for Markov Modulated Queues

Most of the heavy-traffic analysis literature in queueing focuses on systems with i.i.d. interarrival and service times. However, in many applications, this assumption is not valid. In this paper, we extend the Transform Method introduced in Hurtado-Lange and Maguluri (2020) to allow Markov-modulated arrivals and service processes. A key step in this method is carefully designing a test function such that its drift yields the moment-generating function of the vector of queue lengths. However, the generalization to systems where the arrival and service processes may depend on the queue lengths remained open. This paper proposes using the Poisson equation to design this test function instead of an informed guess. Our methodology is clean and we believe has the potential to be extended for a variety of queueing networks. In this paper, we obtain the heavy-traffic distribution of the queue lengths in a load-balancing system under join-the-shortest queue (JSQ) where the arrival and service processes are Markov modulated. In the past, a similar result was known where only the arrival process was Markov-modulated and the service times were i.i.d. exponential.