Time-dependent analysis of M/G/1 vacation models with exhaustive service

We analyze the time-dependent process in severalM/G/1 vacation models, and explicitly obtain the Laplace transform (with respect to an arbitrary point in time) of the joint distribution of server state, queue size, and elapsed time in that state. Exhaustive-serviceM/G/1 systems with multiple vacations, single vacations, an exceptional service time for the first customer in each busy period, and a combination ofN-policy and setup times are considered. The decomposition property in the steady-state joint distribution of the queue size and the remaining service time is demonstrated.     

A discrete queue with double thresholds policy and its application to SVC systems

A discrete time Geo/Geo/1 queue with (m, N)-policy is considered in this paper. There are three operation periods being considered: high speed, low speed service periods and idle periods. With double thresholds policy, the server begins to take a working vacation when the number of customers is below m after a service and there is one customer in the system at least. What’s more, if the system becomes empty after a service, the server will take an ordinary vacation. Otherwise, high speed service continues if the number of customers still exceeds m after a service. At the vacation completion instant, servers resume their service if the quantity of customers exceeds N. Vacations can also be interrupted when the system accumulate customers more than the prefixed threshold. Using the quasi birth-death process and matrix-geometric solution methods, we derive the stationary queue length distribution and some system characteristics of interest. Based on these, we apply the queue to a virtual channel switching system and present various numerical experiments for the system. Finally, numerical results are offered to illustrate the optimal (m, N)-policy to minimize cost function and obtain practical consequence on the operation of double thresholds policy.     

Batch arrival queue with N-policy and at most J vacations

This paper studies the operating characteristics of an M^[x]/G/1 queueing system with N-policy and at most J vacations. The server takes at most J vacations repeatedly until at least N customers returning from a vacation are waiting in the queue. If no customer arrives by the end of the Jth vacation, the server becomes idle in the system until the number of arrivals in the queue reaches N. We derive the system size distribution at a random epoch and departure epoch, as well as various system characteristics.     

Stochastic bounds for a polling system

In this note we consider two queueing systems: a symmetric polling system with gated service at allN queues and with switchover times, and a single-server single-queue model with one arrival stream of ordinary customers andN additional permanently present customers. It is assumed that the combined arrival process at the queues of the polling system coincides with the arrival process of the ordinary customers in the single-queue model, and that the service time and switchover time distributions of the polling model coincide with the service time distributions of the ordinary and permanent customers, respectively, in the single-queue model. A complete equivalence between both models is accomplished by the following queue insertion of arriving customers. In the single-queue model, an arriving ordinary customer occupies with probabilityp _i a position at the end of the queue section behind theith permanent customer,i = l, ...,N. In the cyclic polling model, an arriving customer with probabilityp _i joins the end of theith queue to be visited by the server, measured from its present position.For the single-queue model we prove that, if two queue insertion distributions {p _{i, i} = l, ...,N} and {q _{i, i} = l, ...,N} are stochastically ordered, then also the workload and queue length distributions in the corresponding two single-queue versions are stochastically ordered. This immediately leads to equivalent stochastic orderings in polling models.Finally, the single-queue model with Poisson arrivals andp ₁ = 1 is studied in detail.Part of the research of the first author has been supported by the Esprit BRA project QMIPS.     

Cycle analysis of a two-phase queueing model with threshold

We consider a single-server, two-phase queueing system with N-policy. Customers arrive at the system according to a Poisson process and receive batch service in the first phase followed by individual services in the second phase. If the system becomes empty at the moment of the completion of the second-phase services, it is turned off. After an idle period, when the queue length reaches N (threshold), the server is turned on and begins to serve customers. We obtain the system size distribution and show that the system size decomposes into three random variables. The system sojourn time is provided. Analysis for the gated batch service model is also provided. Finally we derive a condition under which the optimal operating policy is achieved.     

Comparison of two randomized policy M/G/1 queues with second optional service, server breakdown and startup

The problem addressed in this paper is to compare the minimum cost of the two randomized control policies in the M/G/1 queueing system with an unreliable server, a second optional service, and general startup times. All arrived customers demand the first required service, and only some of the arrived customers demand a second optional service. The server needs a startup time before providing the first required service until the system becomes empty. After all customers are served in the queue, the server immediately takes a vacation and the system operates the (T, p)-policy or (p, N)-policy. For those two policies, the expected cost functions are established to determine the joint optimal threshold values of (T, p) and (p, N), respectively. In addition, we obtain the explicit closed form of the joint optimal solutions for those two policies. Based on the minimal cost, we show that the optimal (p, N)-policy indeed outperforms the optimal (T, p)-policy. Numerical examples are also presented for illustrative purposes.     

Some results on a generalized M/G/1 feedback queue with negative customers

This paper deals with a generalized M/G/1 feedback queue in which customers are either “positive" or “negative". We assume that the service time distribution of a positive customer who initiates a busy period is G _e (x) and all subsequent positive customers in the same busy period have service time drawn independently from the distribution G _b (x). The server is idle until a random number N of positive customers accumulate in the queue. Following the arrival of the N-th positive customer, the server serves exhaustively the positive customers in the queue and then a new idle period commences. This queueing system is a generalization of the conventional N-policy queue with N a constant number. Explicit expressions for the probability generating function and mean of the system size of positive customers are obtained under steady-state condition. Various vacation models are discussed as special cases. The effects of various parameters on the mean system size and the probability that the system is empty are also analysed numerically. AMS Subject Classification: Primary: 60 K 25 · Secondary: 60 K 20, 90 B 22     

Workload and Waiting Time Analyses of MAP/G/1 Queue under D-policy

We study the workload (unfinished work) and the waiting time of the queueing system with MAP arrivals under D-policy. The D-policy stipulates that the idle server begin to serve the customers only when the sum of the service times of all waiting customers exceeds some fixed threshold D. We first set up the system equations for workload and obtain the steady-state distributions of workloads at an arbitrary idle and busy points of time. We then proceed to obtain the waiting time distribution of an arbitrary customer based on the workload results. The M/G/1/D-policy queue will be investigated as a special case.     

Dynamic server assignment in a two-queue model

We consider a polling model of two M/G/1 queues, served by a single server. The service policy for this polling model is of threshold type. Service at queue 1 is exhaustive. Service at queue 2 is exhaustive unless the size of queue 1 reaches some level T during a service at queue 2; in the latter case the server switches to queue 1 at the end of that service. Both zero- and nonzero switchover times are considered. We derive exact expressions for the joint queue length distribution at customer departure epochs, and for the steady-state queue-length and sojourn time distributions. In addition, we supply a simple and very accurate approximation for the mean queue lengths, which is suitable for optimization purposes.     

Approximating queue size and waiting time distributions in general polling systems

Polling system models are extensively used to model a large variety of computer and communication networks as well as production and service systems in which multiple customer classes or a number of distinct items compete for the capacity of a common server or production facility. In this paper we describe an efficient approximation method for the steady state distributions of the queue sizes and waiting times. This method is highly accurate as demonstrated by an extensive numerical study. In addition, it is highly adaptable to a variety of arrival patterns and switching protocols, including exhaustive and gated regimes, simple cyclical systems as well as general polling tables. For a system withN stations, one finds the firstK probability density function values of the steady state queue size in any given station inO(max(N, K ²) time only. When executed on an IBM system RS/6000, we have observed an average CPU time of less than 1 second for systems with as many as 50 stations over a large variety of parameter settings.     

Polling and greedy servers on a line

A single server moves with speed on a line interval (or a circle) of length (circumference)L. Customers, requiring service of constant durationb, arrive on the interval (or circle) at random at mean rate customers per unit length per unit time. A customer's mean wait for service depends partly on the rules governing the server's motion. We compare two different servers: thepolling server and thegreedy server. Without knowing the locations of waiting customers, a polling server scans endlessly back and forth across the interval (or clockwise around the circle), stopping only where it encounters a waiting customer. Knowing where customers are waiting, a greedy server always travels toward the current nearest one. Except for certain extreme values of ,L, b, and, the polling and greedy servers are roughly equally effective. Indeed, the simpler polling server is often the better. Theoretical results show, in most cases, that the polling server has a high probability of moving toward the nearest customer, i.e. moving as a greedy server would. The greedy server is difficult to analyze, but was simulated on a computer.     

Bulk input queues with hysteretic control

This paper deals with a bulk input-batch service queueing system and (r,N)-hysteretic control, i.e., under N-policy and r-quorum discipline. The model also includes state dependent service. The goal of such global control is to reduce the waste of server capacity (r-quorum), an unwanted number of switchovers between idle and busy modes (N-policy), and the queue length (by means of variable service rates). The analysis of the system is based on first excess level technique developed by the second author. This approach enables the authors to obtain major characteristics for the queueing process in a closed analytical form. This revised version was published online in June 2006 with corrections to the Cover Date.     

An optimal service rate in a Poisson arrival queue with two-stage service policy*

We consider a two-stage service policy for a Poisson arrival queueing system. The idle server starts to work with ordinary service rate when a customer arrives. If the number of customers in the system reaches N, the service rate gets faster and continues until the system becomes empty. Otherwise, the server finishes the busy period with ordinary service rate. After assigning various operating costs to the system, we show that there exists a unique fast service rate minimizing the long-run average cost per unit time.This work was supported by Korea Research Foundation Grant(KRF-2002-070-C00021).     

Steady-state probability of the randomized server control system with second optional service, server breakdowns and startup

This paper deals with the 〈N,p〉-policy M/G/1 queue with server breakdowns and general startup times, where customers arrive to demand the first essential service and some of them further demand a second optional service. Service times of the first essential service channel are assumed to follow a general distribution and that of the second optional service channel are another general distribution. The server breaks down according to a Poisson process and his repair times obey a general distribution in the first essential service channel and second optional service channel, respectively. The server operation starts only when N (N≥1) customers have accumulated, he requires a startup time before each busy period. When the system becomes empty, turn the server off with probability p (p∈[0,1]) and leave it on with probability (1?p). The method of maximum entropy principle is used to develop the approximate steady-state probability distribution of the queue length in the M/G(G, G)/1 queueing system. A study of the derived approximate results, compared to the established exact results for three different 〈N,p〉-policy queues, suggests that the maximum entropy principle provides a useful method for solving complex queueing systems.     

Ergodicity and analysis of the process describing the system state in polling systems with two queues

Consider a polling system of two queues served by a single server that visits the queues in cyclic order. The polling discipline in each queue is of exhaustive-type, and zero-switchover times are considered. We assume that the arrival times in each queue form a Poisson process and that the service times form sequences of independent and identically distributed random variables, except for the service distribution of the first customer who is served at each polling instant (the time in which the server moves from one queue to the other one). The sufficient and necessary conditions for the ergodicity of such polling system are established as well as the stationary distribution for the continuous-time process describing the state of the system. The proofs rely on the combination of three embedded processes that were previously used in the literature. An important result is that ρ=1 can imply ergodicity in one specific case, where ρ is the typical traffic intensity for polling systems, and ρ<1 is the classical non-saturation condition.     

Admission control for a multi-server queue with abandonment

In a M/M/N+M queue, when there are many customers waiting, it may be preferable to reject a new arrival rather than risk that arrival later abandoning without receiving service. On the other hand, rejecting new arrivals increases the percentage of time servers are idle, which also may not be desirable. We address these trade-offs by considering an admission control problem for a M/M/N+M queue when there are costs associated with customer abandonment, server idleness, and turning away customers. First, we formulate the relevant Markov decision process (MDP), show that the optimal policy is of threshold form, and provide a simple and efficient iterative algorithm that does not presuppose a bounded state space to compute the minimum infinite horizon expected average cost and associated threshold level. Under certain conditions we can guarantee that the algorithm provides an exact optimal solution when it stops; otherwise, the algorithm stops when a provided bound on the optimality gap is reached. Next, we solve the approximating diffusion control problem (DCP) that arises in the Halfin–Whitt many-server limit regime. This allows us to establish that the parameter space has a sharp division. Specifically, there is an optimal solution with a finite threshold level when the cost of an abandonment exceeds the cost of rejecting a customer; otherwise, there is an optimal solution that exercises no control. This analysis also yields a convenient analytic expression for the infinite horizon expected average cost as a function of the threshold level. Finally, we propose a policy for the original system that is based on the DCP solution, and show that this policy is asymptotically optimal. Our extensive numerical study shows that the control that arises from solving the DCP achieves a very similar cost to the control that arises from solving the MDP, even when the number of servers is small.     

Complete characterisation of the customer delay in a queueing system with batch arrivals and batch service

Whereas the buffer content of batch-service queueing systems has been studied extensively, the customer delay has only occasionally been studied. The few papers concerning the customer delay share the common feature that only the moments are calculated explicitly. In addition, none of these surveys consider models including the combination of batch arrivals and a server operating under the full-batch service policy (the server waits to initiate service until he can serve at full capacity). In this paper, we aim for a complete characterisation—i.e., moments and tail probabilities - of the customer delay in a discrete-time queueing system with batch arrivals and a batch server adopting the full-batch service policy. In addition, we demonstrate that the distribution of the number of customer arrivals in an arbitrary slot has a significant impact on the moments and the tail probabilities of the customer delay.     

Two thresholds of a batch arrival queueing system under modified T vacation policy with startup and closedown

This paper studies the operating characteristics of the variant of an M^[x]/G/1 vacation queue with startup and closedown times. After all the customers are served in the system exhaustively, the server shuts down (deactivates) by a closedown time, and then takes at most J vacations of constant time length T repeatedly until at least one customer is found waiting in the queue upon returning from a vacation. If at least one customer is present in the system when the server returns from a vacation, then the server reactivates and requires a startup time before providing the service. On the other hand, if no customers arrive by the end of the J th vacation, the server remains dormant in the system until at least one customer arrives. We will call the vacation policy modified T vacation policy. We derive the steady‐state probability distribution of the system size and the queue waiting time. Other system characteristics are also investigated. The long‐run average cost function per unit time is developed to determine the suitable thresholds of T and J that yield a minimum cost. Copyright © 2007 John Wiley & Sons, Ltd.