Idle Servers Research Articles

Optimal ( d, c) vacation policy for a finite buffer M/M/c queue with unreliable servers and repairs

This paper considers a finite buffer M/M/c queueing system in which servers are unreliable and follow a ( d, c) vacation policy. With such a policy, at a service completion instant, if the number of customers is reduced to c − d ( c > d), the d idle servers together take a vacation (or leave for a random amount of time doing other secondary job). When these d servers return from a vacation and if still no more than c − d customers are in the system, they will leave for another vacation and so on, until they find at least c − d + 1 customers are in the system at a vacation completion instant, and then they return to serve the queue. This study is motivated by the fact that some practical production and inventory systems or call centers can be modeled as this finite-buffer Markovian queue with unreliable servers and ( d, c) vacation policy. Using the Markovian process model, we obtain the stationary distribution of the number of customers in the system numerically. Some cost relationships among several related systems are used to develop a finite search algorithm for the optimal policy ( d, c) which maximizes the long-term average profit. Numerical results are presented to illustrate the usefulness of such a algorithm for examining the effects of system parameters on the optimal policy and its associated average profit.

Two-class M/G/1 queue under workload control

In this paper, we consider a queueing system under the D-policy in which there are two classes of customers. Class-1 customers have a non-preemptive priority over class-2 customers. The idle server begins to serve the customers only when the total workload of the waiting customers (regardless of their classes) exceeds D. We derive the Laplace-Stieltjes transform of the waiting time of each class of customers. We then derive the mean waiting times. Lastly, we compare our analytical results with the simulation estimates for verification purposes.

M/M/3/3 and M/M/4/4 retrial queues

This paper studies M/M/$c$/$c$ retrial queues, where $c$ servers are all identical. In the retrial queues, an arriving customer is served immediately if it finds an idle server upon arrival, otherwise the customer tries to enter the system after an exponentially distributed time independently of other customers. As is well known, it is a challenging problem to obtain an analytical solution for the stationary joint distribution of the numbers of retrial customers and busy servers in the M/M/$c$/$c$ retrial queue especially for $c \ge 3$. Under some technical assumptions, a few analytical solutions have been presented for $c \ge 3$. This paper derives analytical solutions for M/M/3/3 and M/M/4/4 retrial queues without such technical assumptions. Through many numerical examples, we show that the derived analytical solutions can be computed by a numerically stable algorithm.

The BMAP/PH/N retrial queue with Markovian flow of breakdowns

We consider a multi-server retrial queue with the Batch Markovian Arrival Process ( BMAP). The servers are identical and independent of each other. The service time distribution of a customer by a server is of the phase ( PH) type. If a group of primary calls meets idle servers the primary calls occupy the corresponding number of servers. If the number of idle servers is insufficient the rest of calls go to the orbit of unlimited size and repeat their attempts to get service after exponential amount of time independently of each other. Busy servers are subject to breakdowns and repairs. The common flow of breakdowns is the MAP. An event of this flow causes a failure of any busy server with equal probability. When a server fails the repair period starts immediately. This period has PH type distribution and does not depend on the repair time of other broken-down servers and the service time of customers occupying the working servers. A customer whose service was interrupted goes to the orbit with some probability and leaves the system with the supplementary probability. We derive the ergodicity condition and calculate the stationary distribution and the main performance characteristics of the system. Illustrative numerical examples are presented.

A discrete-time retrial queueing model with one server

This paper presents a one-server queueing model with retrials in discrete-time. The number of primary jobs arriving in a time slot follows a general probability distribution and the different numbers of primary arrivals in consecutive time slots are mutually independent. Each job requires from the server a generally distributed number of slots for its service, and the service times of the different jobs are independent. Jobs arriving in a slot can start their service only at the beginning of the next slot. When upon arrival jobs find the server busy all incoming jobs are sent into orbit. When upon arrival in a slot jobs find the server idle, then one of the incoming jobs (randomly chosen) in that slot starts its service at the beginning of the next slot, whereas the other incoming jobs in that slot, if any, are sent into orbit. During each slot jobs in the orbit try to re-enter the system individually, independent of each other, with a given retrial probability. The ergodicity condition and the generating function of the joint equilibrium distribution of the number of jobs in orbit and the residual service time of the job in service are calculated. From the generating function several performance measures are deduced, like the average orbit size. Also the busy period and the number of jobs served during a busy period are discussed. To conclude, extensive numerical results are presented.

Central limit theorem for a many-server queue with random service rates

Given a random variable $N$ with values in ${\mathbb{N}}$, and $N$ i.i.d. positive random variables $\{\mu_k\}$, we consider a queue with renewal arrivals and $N$ exponential servers, where server $k$ serves at rate $\mu_k$, under two work conserving routing schemes. In the first, the service rates $\{\mu_k\}$ need not be known to the router, and each customer to arrive at a time when some servers are idle is routed to the server that has been idle for the longest time (or otherwise it is queued). In the second, the service rates are known to the router, and a customer that arrives to find idle servers is routed to the one whose service rate is greatest. In the many-server heavy traffic regime of Halfin and Whitt, the process that represents the number of customers in the system is shown to converge to a one-dimensional diffusion with a random drift coefficient, where the law of the drift depends on the routing scheme. A related result is also provided for nonrandom environments.

The performance of the M/G/1 queue under the dyadic [formula omitted]-policy and its cost optimization

In this paper, we consider the M/G/1 queueing system under the Min ( N , D ) -policy in which the idle server resumes its service if either N customers accumulate in the system or the total backlog of the service times of the waiting customers exceeds D , whichever occurs first. We first analyze the queue length, the workload and the waiting time. Then, we consider two linear cost models (one based on the mean workload and the other based on the mean queue length) and compare the optimal Min ( N , D ) -policy with the optimal N -policy and the optimal D -policy.

Multi-Server Queues with Intermediate Buffer and Delayed Information on Service Completions

A controller with an unlimited buffer receives messages to be dispatched to c servers downstream in the network. However, the queue sizes at the individual servers are not known exactly, since information on each service completion reaches the controller only after some random delay. The controller can dispatch messages as soon as they arrive, in a cyclic manner, to the c servers. Alternatively, s/he can wait until full information is gained and dispatch a waiting message to a server only when s/he is sure that the server is free. Another strategy is to maintain a limited intermediate buffer in front of the servers, and forward messages to this buffer when information on service completion reaches the controller. If a server completes a job and the intermediate buffer is non-empty, it starts serving a job from this buffer with no delay. Such situations are common in many real life processes (such as passport control procedures, or at large waiting rooms in public offices) where customers wait, in front of c servers, to be served. A customer walks (=delay) to the next idle server when s/he sees her/his “waiting number” flashing on the screen. We analyze this model when the underlying process is the M/M/c queue and the information delay is exponential. We use both: i) probability generating functions of the multi-dimensional state space to calculate the boundary probabilities, and ii) matrix geometric approach to derive the stability condition of the system. We show that the intermediate buffer scheme reduces queue sizes and waiting times. Numerical examples are presented.

Temporal correlations of local network losses

We introduce a continuum model describing data losses in a single node of a packet-switched network (like the Internet) which preserves the discrete nature of the data loss process. By construction, the model has critical behavior with a sharp transition from exponentially small to finite losses with increasing data arrival rate. We show that such a model exhibits strong fluctuations in the loss rate at the critical point and non-Markovian power-law correlations in time, in spite of the Markovian character of the data arrival process. The continuum model allows for rather general incoming data packet distributions and can be naturally generalized to consider the buffer server idleness statistics.

Waiting Time of M/M/c/N Queuing System with Balking, Reneging, and Multiple Synchronous Vacations of Partial Servers

Abstract The policy of synchronous vacations of partial servers is considered in an multi-server queuing system with balking and reneging. Some fixed partial idle servers take synchronous multiple vacations when these servers become idle at an instant that a customer leaves the system. In this article, by considering the two cases that the partial servers are on vacation and not on vacation, the distributions of the conditional waiting time for customers who join the system when all the available servers are busy and eventually acquire service are obtained. Based on these, the distribution of the conditional waiting time of customers given that they join the system when all available servers are busy and eventually acquire service is derived.

Multi-server queueing systems with cooperation of the servers

In this paper, we consider an N-server queueing model with homogeneous servers in which customers arrive according to a stationary Poisson arrival process. The service times are exponentially distributed. Two new customer’s service disciplines assuming simultaneous service of arriving customer by all currently idle servers are discussed. The steady state analysis of the queue length and sojourn time distribution is performed by means of the matrix analytic methods. Numerical examples, which illustrate advantage of introduced disciplines comparing to the classical one, are presented.

Performance of the MAP/G/1 Queue Under the Dyadic Control of Workload and Server Idleness

This paper studies the steady-state queue length process of the MAP/G/1 queue under the dyadic control of the D-policy and multiple server vacations. We derive the probability generating function of the queue length and the mean queue length. We then present computational experiences and compare the MAP queue with the Poisson queue.

RELIABILITY OF A k-OUT-OF-n SYSTEM WITH REPAIR BY A SERVICE STATION ATTENDING A QUEUE WITH POSTPONED WORK

In this paper the reliability of a repairable k-out-of-n system is studied. Repair times of components follow a phase type distribution. In addition, the service facility offers service to external customers which arrive according to a MAP. An external customer, who finds an idle server on its arrival, is immediately selected for service. Otherwise, the external customer joins the queue in a pool of postponed work of infinite capacity with probability 1 if the number of failed components in the system is < M (M ≤ n - k + 1) and if the number of failed components ≥ M it joins the pool with probability γ or leaves the system forever. Repair times of components of the system and that of the external customers have independent phase type distributions. At a service completion epoch if the buffer has less than L customers, a pooled customer is taken for service with probability p, 0 < p < 1 If at a service completion epoch no component of the system is waiting for repair, a pooled customer, if any waiting, is immediately taken for service. We obtain the system state distribution under the condition of stability. A number of performance characteristics are derived. A cost function involving L, M, γ and p is constructed and its behaviour investigated numerically.

An M [X]/G/1 retrial queue with server breakdowns and constant rate of repeated attempts

We consider an M[X]/G/1 retrial queue subject to breakdowns where the retrial time is exponential and independent of the number of customers applying for service. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest joins a retrial group (called orbit) to repeat his request later; otherwise, if the server is busy or down, all customers of the coming batch enter the orbit. It is assumed that the server has a constant failure rate and arbitrary repair time distribution. We study the ergodicity of the embedded Markov chain, its stationary distribution and the joint distribution of the server state and the orbit size in steady-state. The orbit and system size distributions are obtained as well as some performance measures of the system. The stochastic decomposition property and the asymptotic behavior under high rate of retrials are discussed. We also analyse some reliability problems, the k-busy period and the ordinary busy period of our retrial queue. Besides, we give a recursive scheme to compute the distribution of the number of served customers during the k-busy period and the ordinary busy period. The effects of several parameters on the system are analysed numerically.

A discrete-time single-server queue with a modified N -policy

We consider a discrete-time single-server queue where the idle server waits for reaching a level N in the queue size to start a batch service of N messages, although the following arrivals during the busy period receive single services. We find the stationary distributions of the queue and system lengths as well as some performance measures. The vacation and busy periods of the system and the number of messages served during a busy period are also analyzed. The stationary distributions of the time spent waiting in the queue and in the system are studied too. Finally, a total expected cost function is developed to determine the optimal operating N-policy at minimum cost.

Quantifying the performance effects of idle time utilization in multiserver systems

AbstractIn many practical multiserver queueing systems, servers not only serve randomly arriving customers but also work on the secondary jobs with infinite backlog during their idle time. In this paper, we propose a c‐server model with a two‐threshold policy, denoted by (e d), to evaluate the performance of this class of systems. With such a policy, when the number of idle servers has reached d (<c), then e (<d) idle agents will process secondary jobs. These e servers keep working on the secondary jobs until they find waiting customers exist in the system at a secondary job completion instant. Using the matrix analytic method, we obtain the stationary performance measures for evaluating different (e, d) policies. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007.

A Non-Work-Conserving Scheduler to Provide Proportional Delay Differentiated Services and Best Effort Service

The proportional delay differentiation (PDD) model provides consistent packet delay differentiation between classes of service. Currently, the present schedulers performing the PDD model cannot achieve desired delay proportion observed in short timescales under light/moderate load. Thus, we propose a Non-Work-Conserving (NWC) scheduler, which utilizes the pseudo-waiting time for an empty queue and forces each class to compare its priority with those of all other classes. Simulation results reveal that NWC outperforms all current schedulers in achieving the PDD model. However, NWC suspends the server from transmitting packets immediately if an empty class has the maximum priority, resulting in an idle server. Therefore, we further propose two approaches, which will serve a best-effort class during this idle time. Compared with other schedulers, the proposed approaches can provide more predictable and controllable delay proportion, accompanied with satisfactory throughput and average queuing delay.

A low cost tax SOA infrastructure in Grid applications

Many high performance database servers are becoming idle with the tax data being integrated to country tax data centers. We make use of the idle servers to set up a provincial tax Grid based on open Grid service architecture (OGSA). We put forward practical methods to integrate databases, to define and create basic modular Grid services and apply agent to manage Grid services. This technical innovation scheme is of a service-oriented architecture (SOA) and succeeds in averting resources waste. Tests proved that it greatly improves the quality of tax services.

Analysis of multi-server queue with a single vacation ( e, d)-policy

In this paper, we consider a Markovian multi-server queue with a single vacation ( e, d)-policy. In an M/M/c system with such a policy, when d (¡ c) servers become idle at a service completion (or a customer departure) instant, e ( ≤ d) idle servers take a single vacation. After the vacation is completed, these servers return to attend the queue (either serve or stay idle) until the vacation condition is satisfied again. This model is more appropriate for the service or manufacturing systems with servers’ maintenance performed during their idle time. We formulate the system as a quasi-birth-and-death (QBD) process and develop the various stationary performance measures for this system. In addition, we also prove several conditional stochastic decomposition properties.

On the Three Threshold Policy in the Multi-Server Queueing System with Vacations

In this paper, we study an M/M/c queue with a three threshold vacation policy denoted by (e, d, N). With such a policy, the servers keep serving the customers until the number of idle servers reaches d and then e of d servers start taking a vacation together. These e servers keep taking vacations until the number of customers in the system is at least N at a vacation completion instant, then the e servers return to serve the queue again. Using the matrix analytic method, we obtain the stationary performance measures and prove the conditional stochastic decomposition properties for the waiting time and queue length. This model is a generalization of previous multi-server vacation models and offers a useful performance evaluation and system design tool in multi-task server queueing systems.