The M/G/1 queue with negative customers

Abstract

We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer—the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

M/GI/1 queues with services of both positive and negative customers

“…which can be obtained from (2.27). Note that this is in agreement with the result of Harrison and Pitel (1996). In this procedure we need…”

“…The system treated in this paper can be seen as a superposition of two symmetrical RCE-FCFS (removal customers at the end FCFS) M/GI/1 G-queues. One of these queues is the same as the one analyzed by Harrison and Pitel (1996); the other can be considered as such a queue by interchanging the roles of positive and negative customers in the first (positive) queue, and has service time B − for negative customers only. Note that negative or positive arrivals to an empty system form a negative or positive queue, respectively, as ordinary customers.…”

The Geo/Geo/1+1 Queueing System with Negative Customers

“…Continuous time queueing systems with negative customers have been discussed extensively in the past years (see the literature [2][3][4]). In recent years, scholars pay more attention to the study of the discrete-time queueing systems with negative arrivals and obtain some significant results.…”

Diffusion Approximation of State-Dependent G-Networks Under Heavy Traffic

“…Thus, every queue in the system may exert some sort of control over the network through the signals. These models have been extensively studied (some examples include [14], [17], [18], [29], [33], [34], [36], [37], [47], [48], and [52]) and are motivated by a series of practical applications. One of the most successful applications, which was also the initial motivation for G-networks, is neural network modeling [21], [24], [30].…”