Two parallel queues with simultaneous service interruptions and/or disasters

Abstract This paper investigates a discrete-time queueing system, which accommodates two types of customers, named type 1 and type 2. Both customer types have their own dedicated queue and their own dedicated server. The service times of all customers are equal to one time slot. Customers arrive in the system independently from slot to slot, but the numbers of arrivals of both types in any slot are not necessarily independent; their joint probability generating function ( pgf ) is $$A(z_1,z_2)$$ . The system operates in an unreliable environment , whereby two types of random distortions may occur, independently from slot to slot, and with given probabilities: minor breakdowns or major breakdowns. Minor breakdowns cause the temporary unavailability of both servers of the system and are referred to as service interruptions in the paper. Major breakdowns result in the simultaneous removal of all customers from the system at random instants in time and are called disasters . Whereas isolated queues with either service interruptions or disasters have been well-studied in the queueing literature, we believe the joint study of both effects, especially in the context of two parallel queues, is novel. We derive a kernel-type functional equation for the steady-state joint pgf $$U(z_1,z_2)$$ of the numbers of type-1 and type-2 customers in the system. Although solving this equation for arbitrary arrival pgfs $$A(z_1,z_2)$$ seems infeasible, we succeed in finding exact closed-form solutions for various specific classes of arrival pgfs $$A(z_1,z_2)$$ . In doing so, we notice that the stability and the behavior of the system are profoundly different, depending on whether disasters do or do not occur. We illustrate our findings abundantly with specific examples and also retrieve some existing results as special cases of our general results.