On the M/G/1 queue by additional inputs

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.

This paper studies the single-server queueing system with two independent input streams: a GI/G and an M/G stream. A new proof is given of an old result which shows how this system can be transformed into an equivalent ‘single input stream’ GI/G/1 queue, and methods to study that equivalent system numerically are given. As part of the numerical analysis, algorithms are given to compute the moments and the distribution function of busy periods in the M/G/1 queue, and of other related busy periods. Special attention is given to the single-server queue with independent D/G and M/G input streams.This work is to be used in the modeling of real-time computer systems, which can often be described as a single-server queueing system with independent D/G and M/G input streams, see for example Ott (1984b).

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

This paper presents an algorithmic analysis of the busy period for the M/M/c queueing system. By setting the busy period equal to the time interval during which at least one server is busy, we develop a first step analysis which gives the Laplace-Stieltjes transform of the busy period as the solution of a finite system of linear equations. This approach is useful in obtaining a suitable recursive procedure for computing the moments of the length of a busy period and the number of customers served during it. The maximum entropy formalism is then used to analyse what is the influence of a given set of moments on the distribution of the busy period and to estimate the true busy period distribution. Our study supplements a recent work of Daley and Servi (1998) and other studies where the busy period of a multiserver queue follows a different definition, i.e., a busy period is the time interval during which all servers are engaged.

This paper presents an algorithmic analysis of the busy period for the M/M/c queueing system. By setting the busy period equal to the time interval during which at least one server is busy, we develop a first step analysis which gives the Laplace-Stieltjes transform of the busy period as the solution of a finite system of linear equations. This approach is useful in obtaining a suitable recursive procedure for computing the moments of the length of a busy period and the number of customers served during it. The maximum entropy formalism is then used to analyse what is the influence of a given set of moments on the distribution of the busy period and to estimate the true busy period distribution. Our study supplements a recent work of Daley and Servi (1998) and other studies where the busy period of a multiserver queue follows a different definition, i.e., a busy period is the time interval during which all servers are engaged.

There many applications throughout industry and government requiring real-time computing. Any application that monitors and/or controls a process would fit into this category. Some examples are: Nuclear power plants, Steel mills, Space program, etc. General Atomics uses eight real-time computer systems for control and high speed data acquisition required to run the nuclear fusion experiments. Real-Time computing can be defined as the ability to respond to asynchronous external events in a predictable (preferably fast) time frame. Real-Time computer systems are similar to other computers in many ways and may by used for general computing requirements such as Time-Sharing. However special hardware, operating systems and software had to be developed to meet the requirement for real-time computing. Traditionally, real-time computing has been a realm of proprietary operating systems with real-time applications written in FORTRAN and assembly language. In the past, these systems adequately served the needs of the real-time world. Many of these systems that were developed 15 years ago are still being used today. However the real-time world is now changing, demanding new systems to be developed. This paper gives a description of general real-time computer systems and how they differ from other systems. However, the main purpose of this paper is to give a detailed technical description of the hardware and operating systems of an existing proprietary system and a real-time UNIX system. The two real-time computer systems described in detail are Modcomp Classic III/95 with the MAX IV operating system and Modcomp TRI-D 9750 with the REAL/IX.2 operating system.

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.

This dissertation examines an M[x]/G/1 queueing system with a randomized vacation policy and at most J vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1-p. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the Jth vacation, the server becomes idle in the system until at least one customer waiting in the queue. In this dissertation, we investigate the following three queueing systems: Reliable server queueing system, un-reliable server queueing system and un-reliable server with a delayed repair queueing system. For the three systems considered in our dissertation, using the supplementary technique, we develop the system size distribution as well as other important system characteristics, such as the system size distribution at busy period initiation epoch, the queue size distribution at a departure epoch, and the distributions of busy period and idle period, etc. Further, for the un-reliable server we also develop main reliability indices of the presented model. Using the renewal reward theorem, a cost model is constructed to determine the optimal randomized vacation policy. Based on the cost model, a heuristic approach is provided to search the joint optimum values of p and J. Some numerical results are presented for illustrative purpose. Our study presents an extension of the existing vacation queueing model and the analysis of the proposed model will provide a useful performance evaluation tool for more general situations arising in real word.

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.

This paper considers a single-server queueing system with Markov-dependent interarrival times, with special reference to the serial correlation coefficient of the arrival process. The queue size and waiting-time processes are investigated. Both transient and limiting results are given.

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.

Consider a single-server queueing system with a Poisson arrival process at rate λ and positive service requirements independently distributed with common distribution functionB(z) and finite expectationwhere βλ < 1, i.e. anM/G/1 system. When the queue discipline is first come first served, or last come first served without pre-emption, the stationary departure process is Poisson if and only ifG=M(i.e.B(z) = 1 − exp (−z/β)); see (8), (4) and (2). In this paper it is shown that when the queue discipline is last come first served with pre-emption the stationary departure process is Poisson whatever the form ofB(z). The method used is adapted from the approach of Takács (10) and Shanbhag and Tambouratzis (9).

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.

We considered a non-Markovian bulk arrival general bulk service queueing system with state dependent arrivals and multiple vacations in this paper. The arrivals occur in bulk with a rateλ, when the server is busy and with a rateλ0, when the server is on vacation. Using a supplementary variable technique and difference differential equations method we derive the queue size distribution, expected length of idle and busy periods and expected queue size of a Mx/G(a, b)/1 queueing system with state dependent arrivals and multiple vacations. We study the same through simulation using ARENA, a powerful manufacturing system simulation software and obtained some performance measures, which are mandatory for the study and analysis of the said queueing model. The simulated results for a few cases are presented.

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.