We propose a generalization of the classical M/M/1 queue process. The resulting model is derived by applying fractional derivative operators to a system of difference-differential equations. This generalization includes both non-Markovian and Markovian properties which naturally provide greater flexibility in modeling real queue systems than its classical counterpart. Algorithms to simulate M/M...

We derive the performance measures for non-priority customers in a priority single-server queue with two types of Markovian arrival processes (MAPs). One type of customers has preemptive resume (or repeat) priority over the other. We regard the model with two types of MAP arrivals as an MAP/SM/l queue with only non-priority customers arrivals, semi-Markovian service times and some server vacati...

In this paper we consider the MMAP/PH/1 priority queue, both the case of preemptive resume and the case of non-preemptive service. The main idea of the presented analysis procedure is that the sojourn time of the low priority jobs in the preemptive case (and the waiting time distribution in the nonpreemptive case) can be represented by the duration of the busy period of a special Markovian flui...

Markovian service process (MSP) is a model similar to the Markovian arrival process (MAP), where arrivals are replaced with service completions. The MSP can represent various queueing models such as vacation models, N -policy models and exceptional service models. We analyze MAP/MSP/1 queues and obtain a new sort of matrix-type factorization of the vector generating function for the stationary ...

We devise an approximation of the marginal queue length distribution in discrete-time polling systems with batch arrivals and fixed packet sizes. The polling server uses the Bernoulli service discipline and Markovian routing. The 1-limited and exhaustive service disciplines are special cases of the Bernoulli service discipline, and traditional cyclic routing is a special case of Markovian routi...

We treat a single–server vacation queue with queue–length dependent vacation schedules. This subsumes the single–server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch–Markovian arrival pro...

In this paper we establish asymptotics for the basic steady-state distributions in a large class of single-server queues. We consider the waiting time, the workload (virtual waiting time) and the steady-state queue lengths at an arbitrary time, just before an arrival and just after a departure. We start by establishing asymptotics for steady-state distributions of Markov chains of M/GI/1 type. ...

This paper examines a discrete-time queuing system with applications to telecommunications traffic. The arrival process is a particular Markov modulated process which belongs to the class of discrete batched Markovian arrival processes. The server process is a single server deterministic queue. A closed form exact solution is given for the expected queue length and delay. A simple system of equ...

We study fluid limits based on ordinary differential equations (ODEs) for Markovian queueing models where nonexponential service times are fit by appropriate Coxian distributions to match their first and second moments. We focus on a heavy-load regime, whereby the fluid limit of the queue-length process of the nonexponential queue estimates a bottleneck situation. Under this condition, we show ...

Garbage collection is a fundamental component of memory management in several software frameworks. We present a general twodimensional Markovian model of a queue with garbage collection where the input process is Markov-modulated and the memory consumption can be modeled with discretisation. We derive important performance measures (also including garbage collection-related measures like mean g...