Stationary waiting time distribution in the infinite-capacity two-queue single-server resequencing system with HOQ-LIFO-LIFO policy operating in random environment

Consideration is given to the waiting time characteristics of the infinite-capacity single-server system with two queues (high and low priority) and negative customers, operating in entirely Markov-modulated environment. Customers arriving to the system occupy one place in the high priority queue and wait there for service. A negative customer arriving to the system moves one customer from the high to the low priority queue (if there is any) and itself leaves the system without having any further effect on it. It is known that such resequencing of customers during the service process results in hard to analyze delay distributions. When the processes are memoryless in case of HOQ-LIFO-LIFO policy (head of queue customer of the higher priority LIFO queue is moved to low-priority LIFO queue), the variance of the waiting time is invariant with respect to the negative arrival rate. Pursuing deeper understanding of this effect, in this short note the first step is made: it is highlighted that the known methodology, based on Kronecker expansions, can be used to obtain the delay distributions in closed-form in terms of Laplace-Stieltjes transform even in the Markov-modulated environment.