Negative Exponential Service Research Articles

A Research on Active Storage Task Allocation Strategy Based on MMC

the processing capacity of a server is unable to satisfy the parallel processing of mass data and also improves data transmission in storage system. This paper has adopted active storage system to transfer calculation task and management ability to data end, improving the parallel processing capability of application program and reducing the data transmission rate in the system. During processing arriving process, this research has complied with Poisson flow, and the service time for each processor is negative exponential distribution and task service rule is the mixing model of “first come, first served”. Through experiment, it is proved that the data transmission rate within MMC system is reduced by 15% than random allocation of task on average and the system speed-up ratio is improved by 2.1 than average.

Analysis of a shared-memory multiprocessor via a novel queuing model

The study on the delay effects caused by memory interference in a parallel processing system with shared memory was implemented using a Machine Repair queuing Model (MRM). In our model the Repair Server has a constant service time and the parallel operating Job Servers have negative exponential service time distributions. Mean Value Analysis (MVA) was used to solve this particular MRM. However, to apply MVA successfully in this model with heterogeneous servers, the response time equation of the Repair Server has been modified. This modified equation was derived by combining mathematics and heuristics. To check this novel method on its accuracy, comparisons were made with as many as 1200 simulations, having a wide spread of model parameters. The results are extremely good, differences between the analytical results and the simulations are always smaller than 2%.

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.

Analysis of the M/Gi/1 →/M/1 queueing model

An M/GI/1 queueing system is in series with a unit with negative exponential service times and infinite waiting room capacity. We determine a closed form expression for the generating function of the joint queue length distribution in steady state. This result is obtained via the solution of a new type of functional equation in two variables.

An N-Server Cutoff Priority Queue

We consider a multi-priority, nonpreemptive, multi-server queueing system, with Poisson arrivals and negative exponential service times. In order to save available servers for higher priority customers, the system deliberately queues arriving lower priority customers whenever the number of busy servers equals or exceeds a given priority-dependent cutoff number. A queued, priority i customer enters service the instant there are fewer than the respective cutoff number of servers busy and all higher priority queues are empty. We derive several performance measures for the waiting time of the ith priority class: its mean, its second moment, and its distribution (in transforms). We also extend the analysis to systems in which any subset of priority levels may overflow to some other system rather than join infinite capacity queues.

A Note on Product-Form Solution for Queuing Networks with Poisson Arrivals and General Service-Time Distributions with Finite Means

Product form was first introduced by Jackson [Operations Research, 1957; Management Science, 1963] who considered only negative exponential service distributions with queue length dependent service rates. Posner and Bernholtz [Operations Research, 1968] showed that one has product form in the case where there are many classes of customers, service times are exponential with class queue length dependent service rates, and lag times (the time for a customer to travel between two servers is a random variable with continuously differentiable distribution function). Baskett, Chandy, Muntz and Palacios [JACM, 1975] extended these results to networks with multiple classes of customers and service time distributions which have rational Laplace transforms vanishing at ∞. Chandy, Howard and Towsley [JACM, 1977] extended this further to the case where the service time distribution functions are continuously differentiable. They also introduced the concept of station balance and established some important relationships between station balance, local balance and product form. In particular, for the closed queueing network they showed that if each queue in the network satisfies local balance in isolation with Poisson arrivals then one has product form

On a Multichannel Queueing System with Ordered Entry and Heterogeneous Servers

Abstract A multichannel queueing system with ordered entry and no waiting line is considered under the assumption of Poisson arrival and negative exponential service with a different rate in each channel. For the system with two or three heterogeneous servers, the best servers' arrangement is established by interchanging the order of servers to decrease the overflow probability and thereby increase the number of items serviced per unit time. The system considered is a simplified model for a production system with a closed loop conveyor.

Technical Note—Conditional Delays Measured in Events for the M/M/c Queue

In many queuing systems the state of the system is known at each arrival epoch. Given an arrival and given the state of the system, the question of what the probability is that the new arrival will be delayed less than t units of time is often posed. For queuing systems with Poisson arrivals, negative exponential service times and various queue disciplines, this may involve considerable computation. In this paper we develop, for the M/M/c queue, recursive relations for calculating the conditional delays where t is measured in events rather than in units of time. These calculations are often simple to perform, even for some queuing models where the delay in units of time has not yet been obtained in closed form.

A Priority Queueing Model for a Mixture of Two Types of Customers

In this paper a queueing model is developed for a system serving two types of customers. Primary customers with a negative exponential service distribution arrive randomly to a group of servers. When all servers are busy with other primary customers new arrivals are assumed to leave without service. Secondary customers with a general distribution of service times can be served only by servers which are not occupied by a primary customer and join a queue if all servers are busy. Primary customers have preemptive priority and are always served ahead of secondary customers. The buffer containing the service required by the delayed secondary customers empties at a rate proportional to the number of servers not occupied by primary customers. Recursive formulas are derived for the distribution and the moments of the content of the buffer in statistical equilibrium. Numerical examples are presented.

The infinite server queue with semi-Markovian arrivals and negative exponential services

The queue with an infinite number of servers with a semi-Markovian arrival process and with negative exponential service times is studied. The queue length process and the type of the last customer to join the queue before time t are studied jointly, both in continuous and in discrete time. Limiting stationary probabilities are also obtained.

The infinite server queue with semi-Markovian arrivals and negative exponential services

The queue with an infinite number of servers with a semi-Markovian arrival process and with negative exponential service times is studied. The queue length process and the type of the last customer to join the queue before time t are studied jointly, both in continuous and in discrete time. Limiting stationary probabilities are also obtained.

Some applications of the theory of infinite capacity service systems to a single server system with linearly state dependent service

Summary A certain single server queueing system with negative exponential service with mean rate nμ, when the system contains n customers, and Poisson arrivals, is formally equivalent to the infinite capacity system M/M/∞. This equivalence is exploited to yield in a very simple manner results for the single server system which were previously obtained by difficult analysis (see Hadidi (1969)).

Some applications of the theory of infinite capacity service systems to a single server system with linearly state dependent service

SummaryA certain single server queueing system with negative exponential service with mean rate nμ, when the system contains n customers, and Poisson arrivals, is formally equivalent to the infinite capacity system M/M/∞. This equivalence is exploited to yield in a very simple manner results for the single server system which were previously obtained by difficult analysis (see Hadidi (1969)).

A note on Belyaev's limiting distribution of the intervals between losses in an n-server system

Summary The aim of the present note is to give an alternative simpler proof to a result of Belyaev [1], namely that in a loss system of n servers with recurrent input and negative exponential service times the intervals between losses, suitably scaled to have constant mean, tend to a negative exponential distribution as n tends to infinity.

A note on Belyaev's limiting distribution of the intervals between losses in an n-server system

SummaryThe aim of the present note is to give an alternative simpler proof to a result of Belyaev [1], namely that in a loss system of n servers with recurrent input and negative exponential service times the intervals between losses, suitably scaled to have constant mean, tend to a negative exponential distribution as n tends to infinity.

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.

On congestion systems with negative exponential desired service time distributions

On congestion systems with negative exponential desired service time distributions

The queue gi/m/2 with service rate depending on the number of busy servers

The time dependent behaviour of the two server queueing system with recurrent input and negative exponential service times is studied here using certain recurrence relations for the underlying queuelength process. The service times have a varying mean depending on the number of busy servers.

A Queueing system with general moving average input and negative exponential service time

If we think of the input to a queueing system as arising from some process and depending on the history of that process, we might well expect the duration of inter-arrival intervals to depend mostly on the recent history and to a much smaller extent on that which is more remote.

Imbedded Markov chain analysis of single server bulk queues

SummaryIn this paper results from Fluctuation Theory are used to analyse the imbedded Markov chains of two single server bulk-queueing systems, (i)with Poisson arrivals and arbitrary service time distribution and (ii) with arbitrary inter-arrival time distribution and negative exponential service time. The discrete time transition probailities and the equilibrium behaviour of the queue lengths of the systems have been obtained along with distributions concerning the busy periods. From the general results several special cases have been derived.