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Service time fluctuations heavily affect the performance of queueing systems, causing long waiting times and backlogs. Recently, it was shown that when service times are solely determined by the server, service resetting can mitigate the deleterious effects of service time fluctuations and drastically improve queue performance (Bonomo et al.,2022). Yet, in many queueing systems, service times have two independent sources: the intrinsic server slowdown ($S$) and the jobs' inherent size ($X$). In these, so-called $S\&X$ queues (Gardner et al., 2017), service resetting results in a newly drawn server slowdown while the inherent job size remains unchanged. Remarkably, resetting can be useful even then. To show this, we develop a comprehensive theory of $S\&X$ queues with service resetting. We consider cases where the total service time is either a product or a sum of the service slowdown and the jobs' inherent size. For both cases, we derive expressions for the total service time distribution and its mean under a generic service resetting policy. Two prevalent resetting policies are discussed in more detail. We first analyze the constant-rate (Poissonian) resetting policy and derive explicit conditions under which resetting reduces the mean service time and improves queue performance. Next, we consider the sharp (deterministic) resetting policy. While results hold regardless of the arrival process, we dedicate special attention to the $S\&X$-M/G/1 queue with service resetting, and obtain the distribution of the number of jobs in the system and their sojourn time. Our analysis highlights situations where service resetting can be used as an effective tool to improve the performance of $S\&X$ queueing systems. Several examples are given to illustrate our analytical results, which are corroborated using numerical simulations.

In 1987, J.W. Cohen analyzed the so-called Serve the Longest Queue (SLQ) queueing system, where a single server attends two non-symmetric $M/G/1$-type queues, exercising a non-preemptive priority switching policy. Cohen further analyzed in 1998 a non-symmetric 2-queue Markovian system, where newly arriving customers follow the Join the Shortest Queue (JSQ) discipline. The current paper generalizes and extends Cohen's works by studying a combined JSQ-SLQ model, and by broadening the scope of analysis to a non-symmetric 3-queue system, where arriving customers follow the JSQ strategy and a single server exercises the preemptive priority SLQ discipline. The system states' multi-dimensional probability distribution function is derived while applying a non-conventional representation of the underlying process's state-space. The analysis combines both Probability Generating Functions and Matrix Geometric methodologies. It is shown that the joint JSQ-SLQ operating policy achieves extremely well the goal of balancing between queue sizes. This is emphasized when calculating the Gini Index associated with the differences between mean queue sizes: the value of the coefficient is close to zero. Extensive numerical results are presented.

We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process, and service time of each individual customer is drawn from a general probability distribution function. Thus, each of the queues comprising the system is, in isolation, an M/G/$\infty$-type queue. A job that is not completed during a visit will have a new service time requirement sampled from the service-time distribution of the corresponding queue. To the best of our knowledge, this paper is the first in which an M/G/$\infty$-type polling system is analysed. For this polling model, we derive the probability generating function and expected value of the queue lengths, and the Laplace-Stieltjes transform and expected value of the sojourn time of a customer. Moreover, we identify the policy that maximises the throughput of the system per cycle and conclude that under the Hamiltonian-tour approach, the optimal visiting order is \emph{independent} of the number of customers present at the various queues at the start of the cycle., Comment: 19 pages, 4 figures, 34 references

The Asymmetric Simple Inclusion Process (ASIP), a lattice-gas model of unidirectional transport and aggregation, was recently proposed as an `inclusion' counterpart of the Asymmetric Simple Exclusion Process (ASEP). In this paper we present an exact closed-form expression for the probability that a given number of particles occupies a given set of consecutive lattice sites. Our results are expressed in terms of the entries of Catalan's trapezoids --- number arrays which generalize Catalan's numbers and Catalan's triangle. We further prove that the ASIP is asymptotically governed by: (i) an inverse square root law of occupation; (ii) a square root law of fluctuation; and (iii) a Rayleigh law for the distribution of inter-exit times. The universality of these results is discussed., Comment: 24 pages, 6 figures

Two independent Poisson streams of jobs flow into a single-server service system having a limited common buffer that can hold at most one job. If a type-i job (i=1,2) finds the server busy, it is blocked and routed to a separate type-i retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. We study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a boundary value problem. Various performance measures are calculated and numerical results are presented., Comment: N° RR-7999 (2012)