Exponential Service Time Distribution Research Articles

Mean value analysis for multiclass closed queueing network models of flexible manufacturing systems with limited buffers

Flexible Manufacturing Systems (FMS) are modelled as multiple class queueing networks with a central server configuration and finite queue capacities. Under the assumption of exponential service time distribution, an extended Mean Value Analysis (MVA) algorithm is developed for the exact evaluation of performance measures. The accuracy of the proposed algorithm is established by comparison with the results of simulation experiments.

Closed two node priority queueing networks

Analytical results for priority queueing networks are extremely rare. Morris (1981) gave the exact solution for closed two node preemptive priority networks with negative exponential service time distributions which were independent of state. In this paper these results are extended to the case of state dependent service rates. Morris also gave an approximate solution to the system with non-preemptive priority at one node. The exact solution is presented and a comparison given between the exact and approximate solution. Also obtained are the exact solutions for two node systems with priorities reversed, state dependent service parameters and generalised service time distributions. Finally, a matrix geometric solution is obtained for two node systems with pre-emptive priority discipline at one node and any non-batch servicing queue discipline at the other.

Busy period distribution in state-dependent queues

We consider a state-dependent M/M/1 queue in which the arrival rate is a function of the instantaneous unfinished work (work backlog) in the system, and the customer's exponential service time distribution is allowed to depend on the unfinished work in the system at the instant that customer arrived. We obtain asymptotic approximations to both the busy period distributions as well as the residual busy period distribution. Our approximations are valid for systems with a rapid arrival rate and small mean service times.

A preemptive priority queue with a general bulk service rule

In this paper a single server preemptive priority queueing system, consisting of two types of units, with unlimited Poisson inputs and exponential service time distributions, is studied. The higher priority units are served in batches according to a general bulk service rule and they have preemptive priority over lower priority units. Steady state queue length distributions, stability condition and the mean queue lengths are obtained.

On the ordering of tandem queues with exponential servers

We consider tandem queues which have a general arrival process. The queueing system consists ofs(s≧ 2) single-server service stations and the servers have exponential service-time distributions. Firstly we give a new proof for the fact that the departure process does not depend on the particular allocation of the servers to the stations. Secondly, considering the service rates, we prove that the departure process becomes stochastically faster as the homogeneity of the servers increases in the sense of a given condition. It turns out that, given the sum of the service rates, the departure process is stochastically fastest in the case where the servers are homogeneous.

On the ordering of tandem queues with exponential servers

We consider tandem queues which have a general arrival process. The queueing system consists of s (s ≧ 2) single-server service stations and the servers have exponential service-time distributions. Firstly we give a new proof for the fact that the departure process does not depend on the particular allocation of the servers to the stations. Secondly, considering the service rates, we prove that the departure process becomes stochastically faster as the homogeneity of the servers increases in the sense of a given condition. It turns out that, given the sum of the service rates, the departure process is stochastically fastest in the case where the servers are homogeneous.

Heavy-Traffic Approximations for Service Systems With Blocking

This paper develops approximations for the blocking probability and related congestion measures in service systems with s servers, r extra waiting spaces, blocked customers lost, and independent and identically distributed service times that are independent of a general stationary arrival process (the G/GI/s/r model). The approximations are expressed in terms of the normal distribution and the peakedness of the arrival process. They are obtained by applying previous heavy-traffic limit theorems and a conditioning heuristic. There are interesting connections to Hayward's approximation, generalized peakedness, asymptotic expansions for the Erlang loss function, the normal-distribution method, and bounds for the blocking probability. For the case of no extra waiting space, a renewal arrival process and an exponential service-time distribution (the GI/M/s/0 model), a heavy-traffic local limit theorem by A. A. Borovkov implies that the blocking depends on the arrival process only through the first two moments of the renewal interval as the offered load increases. Moreover, in this situation Hayward's approximation is asymptotically correct.

A note on multi-queue scheduling of two tasks

Optimal schedules in a central server model, consisting of a `CPU' server and m `I/O' servers, are obtained for the case of two customers. Each customer has exponential service time distributions with a distinct mean at the CPU server and with the same mean at any I/O servers. The performance measure considered is an arbitrarily weighted sum of the CPU utilization for each customer. It is shown that the optimal CPU and I/O schedules are either one of preemptive priority disciplines depending on certain conditions; the conditions are also obtained. A proof is given to the conjecture made by Spirn on the model and the results of Spirn's are generalized.

?/M/1: On the equilibrium distribution of customer arrivals

Each day a facility commences service at time zero. All customers arriving prior to time T are served during that day. The queuing discipline is First-Come First-Served. Each day, each person in the population chooses whether or not to visit the facility that day. If he decides to visit, he arrives at an instant of time such that his expected waiting time in the queue is minimal. We investigate the arrival rate of customers in equilibrium, where each customer is fully aware of the characteristics of the system. We show that the arrival rate is constant before opening time, but that in general it is not constant between opening and closing time. For the case of exponential distribution of service time, we develop a set of equations from which the equilibrium queue size distribution and expected waiting time can be numerically computed as functions of time.

On an exponential server with general cyclic arrivals

A E n¦ M ¦ 1 queue is defined as a single server queue with exponential service time and general cyclic arrival distributions of cycle length n. The waiting time distribution for such a queue is proved to be a sum of n exponential terms; this is a generalization of G ¦ M ¦ 1 queue results. Based on this a method for obtaining the steady-state waiting time distributions for E n¦ M ¦ 1 queues is introduced. An example is presented to show an application of E n¦ M ¦ 1 queues in deterministic routing.

Statistical analysis of a system for the metrological servicing of measuring devices

An analysis of the MS system operating in the given scientific-research institute has led to the conclusion that it can be mathematically described by a model such that the incoming stream into the system has a Poisson distribution function with exponential service-time distribution. It is evident from the results of the analysis of the system that even the small set of statistical data available in present-day practice for any metrological service is sufficient for the execution of calculations by which it is possible to control the processes of metrological servicing of instruments. It is also quite apparent that without a sufficiently thorough analysis and the application of mathematical statistical methods it is difficult to determine the optimal operating conditions for the MS system and to explore the reserves for better utilization of instruments, increasing the labor performance, improving the planning of fundamental criteria, and increasing the over-all cost effectiveness of the metrological service.

Technical Note—A Transient Solution to the Multistage Poisson Queueing System with Infinite Servers

This paper gives an alternative method of deriving the time dependent solution of a multistage nonqueueing process, with random arrivals into the first stage of the system, sufficient servers at each stage and exponential service time distributions. In the standard notation, the system is a finite number of M/M/∞ queues in series (tandem). This method has the advantage over previous methods of being able to solve the case when any of the stage average service times are equal. The solution was obtained during a project concerned with developing a mathematical model of Acute Myeloid Leukemia patient progress.

Socially and Individually Optimal Control of Arrivals to a GI/M/1 Queue

We study the problem of controlling the arrivals to a single-server queue with general interarrival-time distribution and exponential service-time distribution (a GI/M/1 system). Control is exercised by accepting or rejecting arriving customers or by charging entering customers a toll or entrance fee. An entering customer receives a reward, representing his utility of service. The rewards vary randomly from customer to customer. There is a holding cost, which is a convex function of the number of customers in the system. We compare socially and individually optimal entering policies when the decision horizon is finite, specifically, when the system must close after n future arrivals. There may be a terminal reward or cost, incurred as a function of the number of customers present at the end of the horizon, representing, for example, the holding cost incurred until all of them are served. We show that a socially optimal policy is less likely to accept a customer than an individually optimal policy, and both policies are less likely to accept as the number of customers already present increases, the horizon length increases, or the discount rate decreases. It is shown that, contrary to intuition, the optimal congestion toll cannot be monotonic in the number of customers present. These properties carry over to the infinite horizon, both with and without discounting, except that in the latter case, in which the criterion is long-run average net benefit, the toll can be monotonic. Implications for the study of rush-hour or peak-load phenomena in various congestion systems are briefly discussed.

The interstage buffer storage capacity of non-powered assembly lines A simple mathematical approach

This paper considers the properties of the interstage buffer storage capacity as a ‘ decoupling agent ’ in multi-station non-powered assembly lines, where the economics of production are achieved by balancing delay costs against stock-carrying costs and storage space costs. A simplified mathematical model for exponential service time distribution based on the ‘ cyclic queue ’ concept has been developed, taking into account the three cost parameters, to optimize the buffer storage capacity. A numerical example and sensitivity analysis of the model are presented.

Cyclic Queue Models of Fleet Operations

Fleets of liquid natural gas vessels operating between one loading and one or two discharge ports are an example of vehicles that operate in closed cycles between a small number of terminals. We extend previous work to develop a cyclic queue model that represents such fleet operations. For any specific system we obtain the expected number of vessels in each stage, the expected number waiting in each stage, and, most important, the expected waiting time in port. Exponential service time distributions are assumed, but a series of parallel simulation computations examines the effects of other distributions. The results for exponential service are obtained in closed form, so that the simulation model can be tested and the confidence interval of the simulations estimated. Both analytic and simulation results are presented for the exponential service case under several sets of system parameters. Simulation results for the same sets are presented for normally distributed service and travel times. We restrict the detailed analysis to balanced systems but discuss extensions to more general systems.

Facility Assignment in a Mixed Model Assembly Line

This paper presents an analysis of the problem of Assembly Facility Assignment in a Mixed Model Assembly Line. Poisson arrivals and exponential service time distribution have been assumed. On the basis of the analysis presented it is possible to choose a particular facility on economic considerations, further, optimum service rate for such facilities has also been evaluated.

Computer-controlled queues in series

The paper presents an investigation of a computer-controlled queuing system with N queues in series and Mi, 1 ≤ i ≤ N, servers of negative exponential service-time distribution serving the ith queue. A generating function for stationary-stateprobabilities of necessary and sufficient conditions for the existence of statistical equilibrium for, and a formula for thedetermination of, the waiting-time distribution in the system are obtained.

Queues with Scheduled Arrivals: A Correction, Simplification and Extension

Summary Customers are scheduled to arrive at a queue during equal time intervals but they may be late; the lateness distribution is perfectly general. Bulk arrivals with an exponential service time distribution, single arrivals with staged services and single arrivals with a general service time distribution are considered when customers must arrive in the interval for which they were scheduled or not at all. Results given by the author in a previous paper, for single arrivals who may come in a later interval than the one for which they were scheduled and have an exponential service time distribution, are corrected. It is concluded that when customers are scheduled then the equilibrium distribution of queue length is a truncated sum of weighted geometric distributions, irrespective of the number of arrivals per interval, the lateness distribution, the service time distribution and the number of servers.

Remark on a Service System with Mixed Input Streams

In a paper published in Opns. Res. 14, 145–156 (1966) J. P. Young discusses multiple-channel queuing systems with exponential service-time distribution, no waiting space, and two parallel input streams (Poisson input mixed with an L-phase Erlang input). This note discusses the consequences of an additional assumption introduced by Young to obtain steady-state probabilities of having n channels occupied. Since the assumption is justified only for the pure Poisson input in the case of mixed input, the Erlang formula (4), proposed by Young, gives only an approximation to the exact probabilities.

Two Queues in Parallel

Summary A queueing system consisting of two queues, each with single servers (independent negative exponential service time distributions) and correlated bivariate Poisson input, is discussed. The joint stationary probabilities for the respective numbers of customers in the two queues in equilibrium are found in the case where we have finite waiting rooms of different sizes for each individual queue. For the model where the waiting-room capacities are unlimited, we derive a functional relationship for the joint probability generating function of the equilibrium probabilities. A survey of published research on queues in parallel is also presented.