The departure process from a queueing system

Departure process of a single server queueing system with Markov renewal input and general service time distribution

Transient behaviour of queueing systems with correlated traffic

Sleep modes are important for user equipment to reduce power consumption or to reduce battery reloading. To keep the user equipment connected, it has to be activated for uplink or downlink data transfer which causes overhead and additional delay. Wireless or mobile equipment has to be monitored repeatedly for location updates. These classes of problems can be modeled by queuing systems with modified busy and idle periods. In this paper, we consider a generalized infinite-buffer, single-server queuing systems with activation overhead and under various types of sleep modes during inactive periods. For the special assumption of Markovian arrival processes of data units and generally distributed activation times and vacation (sleep) periods, the models can be analyzed exactly by a mean value analysis method which is based on regenerative stochastic processes, renewal theory and classical queuing theorems. The method allows also for dynamic sleep periods which are dependent on each other as in the case of the standard IEEE 802.16m for wireless equipment. Explicit results are derived for average values of data unit delays, resource utilization and power- saving margins dependent on given parameters for activation times and wakeup signaling frequencies. The models include the special cases of M/G/1 queuing systems with a modified service time of the first arrival initiating a busy period and M/G/1 queuing systems with vacation periods, respectively, which are well known from literature providing general solutions for the state and delay distributions. Finally, all models are extended to the exact analysis of Batch Poisson arrival processes.

A single-server queueing system with stationary Poisson arrival process, finite buffer and impatient customers as a model of TCP Short Transfer is investigated. Service time of a customer by a server has an exponential distribution. If the server is busy and the buffer is full at a customer arrival epoch, the customer may leave the system forever or move to the orbit. Customers staying in the buffer exhibit signs of impatience: they can leave the buffer lying out of the service. Patience time of a customer has an exponential distribution. When this time expires the customer either leaves the system permanently or goes to the orbit of infinite size. Customers staying in the orbit repeat their attempts to get the service later on in a random amount of time. This time is exponentially distributed with the rate depending or independent of the current number of customers in the orbit.Behavior of the system under study is described by two-dimensional asymptotically quasi-Toeplitz Markov chain. Stability conditions and the algorithms for calculating the stationary state distribution of the chain are obtained. Main performance measures of the system are calculated.KeywordsRetrial queueing systemFinite bufferImpatient customersStationary performance measures

Many researchers have studied variants of queueing systems with vacations. Most of them have dealt with M/G/1 systems and have explicitly analyzed some of their performance measures, such as queue length, waiting time, and so on. Recently, studies on queueing systems whose arrival processes are not Poissonian have appeared. We consider a single server queueing system with multiple vacations and E-limited service discipline, where messages arrive to the system according to a switched Poisson process. First, we consider the joint probability density functions of the queue length and the elapsed service time or the elapsed vacation time. We derive the equations for these pdf's, which include a finite number of unknown values. Using Rouche's theorem, we determine the values from boundary conditions. Finally, we derive the transform of the stationary queue length distribution explicitly.

We consider queueing systems in which the service rate is the controlled variable. The cost depends on the queue length and selected rate. The objective is to choose the service rate dynamically, based on the state of the system so as to minimize the average cost over an infinite horizon. Six classes, either known in the literature or new, are studied in detail: queueing systems with vacations, single-server queueing systems with and without switching costs, and tandem queueing systems with and without service costs. A novel approach is presented here using fuzzy control to solve these problems. Simulation shows that the approach is efficient and promising, especially in cases where analytical solutions do not exist.

System delay versus system content for discrete-time queueing systems subject to server interruptions

In this paper, we consider two queueing models. Model I is on a single-server queueing system in which the arrival process follows MAP with representation D = (D0,D1) of order m and service time follows phase-type distribution (β, S) of order n. When a customer enters into service, a generalized Erlang clock is started simultaneously. The clock has k stages. The pth stage parameter is θp for 1 ≤ p ≤ k. If a customer completes the service in between the realizations of stages k1 and k2 (1 < k1 < k2 < k) of the clock, it is a perfect one. On the other hand, if the service gets completed either before the kth1 stage realization or after the kth2 stage realization, it is discarded because of imperfection. We analyse this model using the matrix-geometric method. We obtain the expected service time and expected waiting time of a tagged customer. Additional performance measures are also computed. We construct a revenue function and numerically analyse it. In Model II, a single server queueing system in which all assumptions are the same as in Model I except the assumption on service time, is considered. Up to stage k1 service time follows phase-type distribution (α′ , T′) of order n1 and beyond stage k1, the service time follows phase type distribution (β′ , S′) of order n2. We compare the values of the revenue function of the two models

In the chapter we describe the application of firefly algorithm in discrete optimization of simple queueing structures such as queueing systems. The optimization of these systems is complicated and there is not any universal method to solve such problem. We briefly cover basic queueing systems. Hence, Markovian systems with exponential service times and a Poisson arrival process with losses, with finite capacity and impatient customers and closed queueing system with finite number of jobs are presented. We consider structural optimization, for example maximization of overall profits and minimizing costs controlled by the number of servers. We show the results of performed experiments.

A single-server queueing system BMAP/SM/1 is studied. It operates as a retrial queueing system under decentralized and centralized retrial strategies with and without loss of primary customers, as well as a system with waiting. The operation mechanism changes under the action of a random environment. The stationary state probability distributions at imbedded and arbitrary instants and main performance characteristics of the system are determined.

We find an equilibrium in a single-server queueing system with retrials and strategic timing of the customers. We consider a set of customers, each of which must decide when to arrive to a queueing system during a fixed period of time. In this system, after completion of service, the server seeks a customer blocked in a virtual orbit (orbital customer) to be served next, unless a new customer captures the server. We develop, in detail, a setting with two and three customers in the set, and formulate and discuss the problem for the general case with an arbitrary number of customers. The numerical examples for the system with two and three customers included as well.

To analyze networks of queues, it is important to be able to analyze departure processes from single queues. For the M/M/s and M/G/∞ models, the stationary departure process is simple (Poisson), but in general the stationary departure process is quite complicated. As a basis for approximations, this paper shows that the stationary departure process is approximately Poisson when there are many busy slow servers in a large class of stationary G/GI/s congestion models having s servers, infinite waiting room, the first-come first-served discipline, and mutually independent and identically distributed service times that are independent of a stationary arrival process. Limit theorems are proved for the departure process in a G/GI/s system in which the number of servers and the offered load (arrival rate divided by the service rate) both increase. The asymptotic behavior of the departure process depends on the way the arrival rate changes. If the arrival rate is held fixed, so that the offered load increases by slowing down the service rate, then the departure process converges to a Poisson process. For this result, the service-time distribution is assumed to be phase-type. Other limiting behavior occurs if the arrival rate approaches zero or infinity. Convergence is established in each case by applying previous heavy-traffic limit theorems.

Clarifying methods of Truini et al. [Pain 2007;131:343–7] and proposing further evidence supporting the “first come first served” hypothesis: A reply to Mouraux and Iannetti

CPU scheduling is the basis of multi-programmed operating systems. The scheduler is accountable for multiplexing processes on the CPU. By switching the CPU among processes, the operating system results in the computer more prolific. Various CPU scheduling algorithms exist for a multi-programmed operating system like First Come First Served (FCFS), Shortest Job First (SJF), Shortest Remaining Time Next (SRTN) and Round Robin (RR) scheduling algorithm. This paper has threefold objective. Firstly, it throws light on a new-fangled CPU scheduling algorithm called MIN-MAX whichbehaves as both preemptive and non-preemptive algorithm basis on the burst time. Secondly, it would helps to understand how MIN-MAXalgorithm improves the CPU efficiency in multi programming operating system and also trims down the starvation problem amongprocesses. Thirdly, it focuses on the comparative study of the existing algorithms on basis of various scheduling parameters with the proposed algorithm MIN-MAX.

In this paper we assume that the customers arrive at a single server queueing system according to Markovian Arrival process. When the system is empty, the server goes for vacation and produces inventory for future use during this period. The maximum processed inventory at a stretch is L. The inventory processing time follows phase type distribution. These are required for the service of customers-one for each customer. The server returns from vacation when there are N customers in the system. The service time follows two distinct phase type distributions depending on whether there is processed items or no processed item available at service commencement epoch. We analyse the distributions of time till the number of customers hit N or the inventory level reaches L, idle time, the distribution of time until the number of customers hit N and also the distribution of the number of inventory processed before the arrival of first customer. Also we provide the distribution of a busy cycle, LSTs of busy cycles in which no item is left in the inventory and at least one item is left in the inventory. We perform some numerical experiments to evaluate the expected idle time, standard deviation and coefficient of variation of idle time of the server.