Idle Servers Research Articles

Multiple anticipatory moves of a server on a line

This paper investigates the destination and desirability of anticipatory moves by an idle server aimed at reducing its response time for future requests. It is motivated by certain types of automated storage and retrieval systems, computer disks and elevators. Requests' timing and location are assumed to be uncertain. If a request arrives during an anticipatory move, the server will travel to the request upon the completion of the noninterruptible move. Otherwise, another anticipatory move can be considered in order to be closer to the static optimum. A stochastic dynamic programming model is developed for finding whether and where to move from each starting location. Structural properties of the optimal solutions are analyzed. Numerical examples are explored to provide insight into the nature and benefit of such moves. These numerical examples show the benefits depend heavily on requests' arrival rate and their location distribution, as well as on the server's setup time and speed.

Heterogeneous Multiserver Queues With General Input

A queueing system with heterogeneous servers and general input is analyzed. In particular, it is shown how to obtain the steady state probabilities for such a system, both at random-times and at the times preceding an arrival. If more than one server is idle, an arrival has the choice how to select one of the idle servers. The rules used to do the selection are part of the model description. The probabilities at arrival points are found as follows. First, one determines the probabilities for all states in which there is no queue, and then one calculates the probabilities in which customers are waiting. The probabilities at arrival epochs allow one to find the probability of no wait, the waiting time distribution, and the distribution of the number in the system at random epochs. Heuristic arguments, supported by numerical calculations, show that the importance of selecting a server increases with decreasing traffic intensity, increasing number of servers, and decreasing variation between interarrivai times.

On the stability of retrial queues

We consider the following type of problems. Calls arrive at a queue of capacity K (which is called the primary queue), and attempt to get served by a single server. If upon arrival, the queue is full and the server is busy, the new arriving call moves into an infinite capacity orbit, from which it makes new attempts to reach the primary queue, until it finds it non-full (or it finds the server idle). If the queue is not full upon arrival, then the call (customer) waits in line, and will be served according to the FIFO order. If \lambda is the arrival rate (average number per time unit) of calls and \mu is one over the expected service time in the facility, it is well known that \mu>\lambda is not always sufficient for stability. The aim of this paper is to provide general conditions under which it is a sufficient condition. In particular, (i) we derive conditions for Harris ergodicity and obtain bounds for the rate of convergence to the steady state and large deviations results, in the case that the inter-arrival times, retrial times and service times are independent i.i.d. sequences and the retrial times are exponentially distributeds (ii) we establish conditions for strong coupling convergence to a stationary regime when either service times are general stationary ergodic (no independence assumption), and inter-arrival and retrial times are i.i.d. exponentially distributeds or when inter-arrival times are general stationary ergodic, and service and retrial times are i.i.d. exponentially distributeds (iii) we obtain conditions for the existence of uniform exponential bounds of the queue length process under some rather broad conditions on the retrial process. We finally present conditions for boundedness in distribution for the case of nonpatient (or non persistent) customers.

Analysis of bounded time warp and comparison with YAWNS

This article studies an analytic model of parallel discrete-event simulation, comparing the YAWNS conservative synchronization protocol with Bounded Time Warp. The assumed simulation problem is a heavily loaded queuing network where the probability of an idle server is closed to zero. We model workload and job routing in standard ways, then develop and validate methods for computing approximated performance measures as a function of the degree of optimism allowed, overhead costs of state-saving, rollback, and barrier synchronization, and workload aggregation. We find that Bounded Time Warp is superior when the number of servers per physical processor is low (i.e., sparse load), but that aggregating workload improves YAWNS relative performance.

An approximation method for the M/G/1 retrial queue with general retrial times

In this paper, we consider a single-server queueing system with no waiting space. Customers arrive according to a Poisson process with rate λ. An arriving customer receives immediate service if he finds the server idle; otherwise he will retry for service after a certain amount of time. Blocked customers will repeatedly retry for service until they get served. The times between consecutive attempts are assumed to be independently distributed with common distribution function T(·). The service times of customers are drawn from a common distribution function B(·). We show that the number of customers in the system in steady-state can be decomposed into two independent random variables: the number of customers in the corresponding ordinary M/G/1 queue (with unlimited waiting space) and the number of customers in the retrial queue given that the server is idle. Applying this decomposition property, an approximation method for the calculation of the steady-state queue size distribution is proposed and some properties of the approximation are discussed. It is demonstrated through numerical results that the approximation works very well for models of practical interest.

Optimal Dynamic Assignment of Customers to Heterogeneous Servers in Parallel

The system under consideration comprises two classes of customers to be served by two stations, with parallel servers in each station. While class-1 customers can only receive service from station 1, class-2 customers can be served by either station. Arrival processes of customers form two mutually independent Poisson processes. The service time of a customer at either station is exponentially distributed with a common rate. A class-i customer, while present in the system, will incur a holding cost hi with h1 ≥ h2. The objective is to dynamically assign customers to idle servers so that the expected discounted (or the long-run average) holding cost is minimized. We show that a class-j customer should be assigned to an idle server in station j, j = 1, 2, whenever possible, and a class-2 customer should be assigned to an idle server in station 1 only if (no class-1 customers are waiting, and) the length of queue 2 exceeds a critical number. Moreover, the critical number is monotonically increasing in the number of busy servers in station 1. The numerical results for some test cases are reported.

The idle server first routing strategy in a system with multiple unequal‐rate synchronous servers

Abstract The idle server first (ISF) routing strategy in a finite queueing system with multiple unequal‐rate synchronous servers is studied. This paper extends the work of [4] by considering unequal‐rate servers. The input is assumed to be a batch Poisson distribution. The state transition equations used to describe the queueing behavior of the system are successfully solved. The performance measures, including average message blocking probability, average packet delay, average message delay, and system throughput, are obtained. The results show that ISF routing can have pretty good performance, and its effective measures are almost independent of routing probabilities. Validity of the analysis has also been verified by computer simulations.

Further results for the M/M(a, ∞)/N batch-service system

A multi-server Markovian queueing system is considered such that an idle server will take the entire batch of waiting customers into service as soon as their number is as large as some control limit. Some new results are derived. These include the distribution of the time interval between two consecutive commencements of service (including itsrth moment) and the actual service batch size distribution. In addition, the average customer waiting time in the queue is derived by a simple combinatorial approach.

Throughput maximization in a loss queueing system with heterogeneous servers

Assigning each arriving customer to the fastest idle server is shown to maximize throughput (equivalently, minimize blocking probability) in a queueing model with Poisson arrivals, heterogeneous exponential servers, and no waiting room. If a cost structure is imposed on this model, under specified conditions the same policy minimizes the expected discounted cost and the long-run average cost per unit time.

Throughput maximization in a loss queueing system with heterogeneous servers

Assigning each arriving customer to the fastest idle server is shown to maximize throughput (equivalently, minimize blocking probability) in a queueing model with Poisson arrivals, heterogeneous exponential servers, and no waiting room. If a cost structure is imposed on this model, under specified conditions the same policy minimizes the expected discounted cost and the long-run average cost per unit time.

Heterogeneous-server loss systems with ordered entry: An anomaly

We consider a loss system with Poisson arrivals and ordered entry, and we suppose that server i has service-time distribution function F i . Consistent with the results of others, we verify that, in contrast with the classical Erlang loss system ( F i = F for all i), when F i ≠ F j then the loss probability is not insensitive to the form of F i or Fj (even when F i and F j have the same first moment). Our main point is to note a remarkable counterintuitive fact that is an easy consequence of this result: the policy of assigning arrivals to the fastest idle server does not necessarily produce the lowest loss probability.

Optimal control of a queueing system with simultaneous service requirements

We consider a system with N identical servers, operating in parallel. There are two types of customers, ordinary and locking, having exponential service times with rates μ O and μ L and completion rewards r O and r L (Problem P1) or completion reward r O and holding cost h L (Problem P2), respectively. An ordinary customer needs a single server to be processed, and there are an infinite number of ordinary customers available in the system. Locking customers arrive in a Poisson process with rate λ and require processing by all N servers simultaneously. The servers are allocated in a nonpreemptive manner to both classes. The scheduling decision in such a system consists of determining how the N servers should be allocated, so that the discounted reward, as well as the long run average reward in P1 and P2, is maximized. We prove that the optimal policy is of the following simple form: Have all N servers serve ordinary customers until the queue of the locking customers builds up to some threshold number k*. Then empty the ordinary customers from the N servers as fast as possible by keeping each server idle after it completes the ordinary customer currently in service. Finally, serve all available locking customers, and continue by repeating the above procedure.

The Effect of Idle Server First Random Routing on the Behavior of a Finite Queue

This paper studies the effect of routing on the behavior of a finite queue which accepts batch Poisson inputs and receives service from multiple synchronous servers. Upon the arrival of a group of customers idle or pseudoidle servers will be considered first in routing decisions. Otherwise routing will be determined randomly in accordance with a preset probability distribution. Results obtained include state probability, blocking probability, delay, and throughput. Validity of analysis has been verified by computer simulations. These results can he used in evaluating the performance of a computer communication network.

The value of an information system for service networks

A comparative analysis of various information systems and their corresponding dispatching policies for service networks is presented in this paper. Three information systems are examined 1. (1) Stationary Information System, where only stationary servers at their home locations may be connected to the dispatching center 2. (2) Check-in Information System, where each network node is connected to the center (e.g., designated “call-in” telephone boxes) 3. (3) Real-time Information System, where each server can continuously communicate with the center and consequently be dispatched (e.g., via radio). The latter two systems enable a dispatching policy in which the dispatcher may opt to assign a moving server to a new service request, rather than utilizing only the idle servers. The case when one server is moving while the others are stationary is quantitatively formulated and analysed. The analysis terminates as soon as one of the servers is dispatched. The implications of various information systems and policies are measured mainly by the expected response time to a service request, but the variance of response time is also considered. The gross benefits are calculated for a particular example. Net Profits, Cost-Benefit and Cost-Effectiveness Analyses are the suggested approaches to net economic benefits comparisons of alternative systems, depending on systems' characteristics.

The M/G/k loss system with servers subject to breakdown

This paper studies the equilibrium behaviour of the M/G/k loss system with servers subject to breakdown. Such a system has k servers, whose customers arrive in a Poisson process. They are served if there is an idle server, otherwise they leave and do not return. Each server is subject to breakdown with probability of occurrence depending on the length of the time the server has been busy since his last repair. On a breakdown the customer waits and his service is continued just after the repair of the server. Among other things, a generalization of the Erlang B-formula is given and it is shown that the equilibrium departure process is Poisson. In fact these results are obtained for the more general case where customers may balk and service and repair rates are state dependent.

The M/G/k loss system with servers subject to breakdown

This paper studies the equilibrium behaviour of the M/G/k loss system with servers subject to breakdown. Such a system has k servers, whose customers arrive in a Poisson process. They are served if there is an idle server, otherwise they leave and do not return. Each server is subject to breakdown with probability of occurrence depending on the length of the time the server has been busy since his last repair. On a breakdown the customer waits and his service is continued just after the repair of the server. Among other things, a generalization of the Erlang B-formula is given and it is shown that the equilibrium departure process is Poisson. In fact these results are obtained for the more general case where customers may balk and service and repair rates are state dependent.

The M/G/k group-arrival group-departure loss system

This paper considers the equilibrium behaviour of the M/G/k group-arrival group-departure loss system. Such a system has k servers whose customers arrive in groups, the arrival epochs of groups being points of a Poisson process. The duration of a service can be characteristic of the group size; however, customers who belong to the same group have equal service times. The customers of a group start being served immediately upon their arrival, unless their number is greater than the number of idle servers. In this case the whole group leaves and does not return later (i.e. is lost). Among other things, a generalization of the Erlang B-formula is given and it is shown that the arrival and departure processes are statistically indistinguishable.

The M/G/k group-arrival group-departure loss system

This paper considers the equilibrium behaviour of the M/G/k group-arrival group-departure loss system. Such a system has k servers whose customers arrive in groups, the arrival epochs of groups being points of a Poisson process. The duration of a service can be characteristic of the group size; however, customers who belong to the same group have equal service times. The customers of a group start being served immediately upon their arrival, unless their number is greater than the number of idle servers. In this case the whole group leaves and does not return later (i.e. is lost). Among other things, a generalization of the Erlang B-formula is given and it is shown that the arrival and departure processes are statistically indistinguishable.

Algorithms for the State Probabilities in a General Class of Single Server Queueing Systems with Group Arrivals

Single server queues with batch arrivals are useful in the modeling of computer and communication systems amongst others. In this paper we give for a wide class of single server queues with batch arrivals a stable recursive method to compute state probabilities both at arrival and at arbitrary epochs. This class includes finite capacity models with state dependent Markovian input in which batches could be totally or partially rejected. The customers of a batch are served individually and the service time has a general distribution function. In particular, we discuss as application the computation of operating characteristics, such as rejection probability and probability of server idleness, for a finite capacity M/G/1 queue for which arriving customers are rejected if their waiting plus service times exceed the capacity of the system.

The queuing system M/ GB/ l and its ramifications

This paper deals with the bulk service queuing system M/ G B /1 and its ramifications. In the system M/ G B /1, customers arrive singly by a Poisson process and are served by a single server in order of arrival in batches of fixed maximum size B. The distribution of service time is general and an idle server begins service immediately if one or more customers await service. The main aims of the paper are: 1. (i) to generalize Jaiswal's queuing system M/ E R B /1; 2. (ii) to unify Bailey's and Jaiswal's generalized results and obtain relationships between the pgf's (probability generating functions) P q(z)= ∑ n=0 ∞ P n,1 z n and P +(z)= ∑ n=0 ∞ P n z n + where P n,1 and P n + are the limiting probabilities of n customers in the queue (at random epoch) with server being busy and n in the queue immediately after a departure epoch respectively; 3. (iii) to obtain a simple derivation of the transform of the distribution of waiting time first obtained by Downton and later by others, and to suggest ways of getting computational results in the case when expected values are considered at random epochs rather than at departure epochs as discussed by Bailey and Downton. The supplementary variable technique is used in discussing the above results. It is also demonstrated that Little's formula is true for M/ G B /1 as it should, since it is independent of the distributions involved. Some other operational parameters of this are also discussed. Examples of extensive numerical results for P 0 + and L + are presented in tabular form.