Proof of stability conditions for token passing rings by Lyapunovfunctions

A token passing ring can be described as a system of M queues with one server that rotates around the queues sequentially. Georgiadis-Szpankowski (1992) considered rings where the token (server) performs x ∇ l_j services on queue j, where x is the size of queue j upon arrival of the token, and l_j is a fixed limit of service for queue j. The token then spends some random time switching to the next queue. For j=1, ..., M, arrivals to queue j are Poisson with rate λ_j, and service times have mean s _j and are independent of the arrival and switchover processes. The purpose of this paper is to give an alternate and simpler proof of the stability conditions given by Georgiadis-Szpankowski using Lyapunov functions. An additional assumption is made about the second moments of the service and switchover times being finite     

Stationary distributions and optimal control of queues with batch Markovian arrival process under multiple adaptive vacations

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple adaptive vacation policy. That is, the server serves until system emptied and after that server takes a random maximum number H different vacations until either he finds at least one customer in queue or the server have exhaustively taken all the vacations. The maximum number H of vacations taken by the server is a discrete random variable. We obtain queue-length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, post-departure and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Later we use supplementary variable method and simple algebraic manipulations to obtain the queue-length distribution at other epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue lengths and mean waiting times have been obtained. Several other vacation queueing models can be obtained as a special case of our model, e.g., single-, multiple-vacation model and queues with exceptional first vacation time. Finally, the total expected cost function per unit time is considered to determine a locally optimal multiple adaptive vacation policy at a minimum cost.     

M/M/1-PS queue and size-aware task assignment

We consider a distributed server system in which heterogeneous servers operate under the processor sharing (PS) discipline. Exponentially distributed jobs arrive to a dispatcher, which assigns each task to one of the servers. In the so-called size-aware system, the dispatcher is assumed to know the remaining service requirements of some or all of the existing jobs in each server. The aim is to minimize the mean sojourn time, i.e., the mean response time. To this end, we first analyze an M/M/1-PS queue in the framework of Markov decision processes, and derive the so-called size-aware relative value of state, which sums up the deviation from the average rate at which sojourn times are accumulated in the infinite time horizon. This task turns out to be non-trivial. The exact analysis yields an infinite system of first order differential equations, for which an explicit solution is derived. The relative values are then utilized to develop efficient dispatching policies by means of the first policy iteration (FPI). Numerically, we show that for the exponentially distributed job sizes the myopic approach, ignoring the future arrivals, yields an efficient and robust policy when compared to other heuristics. However, in the case of highly asymmetric service rates, an FPI based policy outperforms it. Additionally, the size-aware relative value of an M/G/1-PS queue is shown to be sensitive with respect to the form of job size distribution, and indeed, the numerical experiments with constant job sizes confirm that the optimal decision depends on the job size distribution.     

The interdeparture time distribution for each class in the ∑Mi/Gi/1 queue with set-up times and repeated server vacations

This paper studies the interdeparture time distribution of one class of customers who arrive at a single server queue where customers of several classes are served and where the server takes a vacation whenever the system becomes empty or is empty when the server returns from a vacation. Furthermore, the first customer in the busy period is allowed to have an exceptional service time (set-up time), depending on the class to which this customer belongs. Batches of customers of each class arrive according to independent Poisson processes and compete with each other on a FIFO basis. All customers who belong to the same class are served according to a common generally distributed service time. Service times, batch sizes and the arrival process are all assumed to be mutually independent. Successive vacation times of the server form independent and identically distributed sequences with a general distribution.For this queueing model we obtain the Laplace transform of the interdeparture time distribution for each class of customers whose batch size is geometrically distributed. No explicit assumptions of the batch size distributions of the other classes of customers are necessary to obtain the results.The paper ends by showing how the mathematical results can be used to evaluate a protocol that controls access to a shared medium of an ATM passive optical network. The numerical results presented in the last section of this paper show that the bundle spacing principle that is used by the permit distribution algorithm of this protocol introduces high delays and in many cases also more variable interdeparture times for the ATM cells of individual connections. An alternative algorithm is proposed that does not cope with these performance short comings and at the same time conserves the good properties of the protocol.     

A Two-Queue Polling Model with Two Priority Levels in the First Queue

In this paper we consider a single-server cyclic polling system consisting of two queues. Between visits to successive queues, the server is delayed by a random switch-over time. Two types of customers arrive at the first queue: high and low priority customers. For this situation the following service disciplines are considered: gated, globally gated, and exhaustive. We study the cycle time distribution, the waiting times for each customer type, the joint queue length distribution at polling epochs, and the steady-state marginal queue length distributions for each customer type.     

Optimal control of a queueing system with two heterogeneous servers

The problem considered is that of optimally controlling a queueing system which consists of a common buffer or queue served by two servers. The arrivals to the buffer are Poisson and the servers are both exponential, but with different mean service times. It is shown that the optimal policy which minimizes the mean sojourn time of customers in the system is of threshold type. The faster server should be fed a customer from the buffer whenever it becomes available for service, but the slower server should be utilized if and only if the queue length exceeds a readily computed threshold value.     

Performance analysis of Markovian polling systems with single buffers

In this paper, we analyze the Markovian polling system with single buffers, asymmetric arrival rates, service times, and switchover times. A virtual buffer model is introduced to derive the relationship of the joint generating function for the queue length of each station at a polling instant. The Laplace-Stieltjes transforms of the cycle time and the intervisit time are obtained from the marginal generating function. We analyze the cyclic, load-oriented-priority, and symmetric random polling schemes which are classified by adjusting the transition probabilities, and compare the merits and demerits of each scheme for the performance measures. In particular, we prove that the mean queue lengths at the polling instants are the same for all stations in case of the load-oriented-priority polling scheme for the buffer relaxation system in which a new message is stored as soon as the transmission of the message currently in the buffer is initiated.     

Analysis of the M/G/1 queue with discriminatory random order service policy

We consider an M/G/1 queue with different classes of customers and discriminatory random order service (DROS) discipline. The DROS discipline generalizes the random order service (ROS) discipline: when the server selects a customer to serve, all customers waiting in the system have the same selection probability under ROS discipline, whereas customers belonging to different classes may have different selection probabilities under DROS discipline. For the M/G/1 queue with DROS discipline, we derive equations for the joint queue length distributions and for the waiting time distributions of each class. We also obtain the moments of the queue lengths and the waiting time of each class. Numerical results are given to illustrate our results.     

On the analysis of a symmetric polling system with single-message buffers

A performance analysis is provided for a polling system consisting of statistically identical stations with single-message buffers and Poisson arrival streams. Switchover and message service times are assumed to be generally distributed. Some errors in the past analysis are pointed out. We express such performance measures as the mean polling cycle time, the mean message response time, and the mean number of messages at an arbitrary time in terms of the mean number of massages served in a polling cycle. Our mean message response time reduces to that for an FCFS M/G/1//N queue (machine interference model) in the limit of zero switchover time.     

On a batch arrival Poisson queue with a random setup time and vacation period

The single server queue with vacation has been extended to include several types of extensions and generalisations, to which attention has been paid by several researchers (e.g. see Doshi, B. T., Single server queues with vacations — a servey. Queueing Systems, 1986, 1, 29–66; Takagi, H., Queueing Analysis: A Foundation of Performance evaluation, Vol. 1, Vacation and Priority systems, Part. 1. North Holland, Amsterdam, 1991; Medhi, J., Extensions and generalizations of the classical single server queueing system with Poisson input. J. Ass. Sci. Soc., 1994, 36, 35–41, etc.). The interest in such types of queues have been further enhanced in resent years because of their theoretical structures as well as their application in many real life situations such as computer, telecommunication, airline scheduling as well as production/inventory systems. This paper concerns the model building of such a production/inventory system, where machine undergoes extra operation (such as machine repair, preventive maintenance, gearing up machinery, etc.) before the processing of raw material is to be started. To be realistic, we also assume that raw materials arrive in batch. This production system can be formulated as an M^x/M/1 queues with a setup time. Further, from the utility point of view of idle time this model can also be formulated as a case of multiple vacation model, where vacation begins at the end of each busy period. Besides, the production/inventory systems, such a model is generally fitted to airline scheduling problems also. In this paper an attempt has been made to study the steady state behavior of such an M^x/M/1 queueing system with a view to provide some system performance measures, which lead to remarkable simplification when solving other similar types of queueing models.This paper deals with the steady state behaviour of a single server batch arrival Poisson queue with a random setup time and a vacation period. The service of the first customer in each busy period is preceded by a random setup period, on completion of which service starts. As soon as the system becomes empty the server goes on vacation for a random length of time. On return from vacation, if he finds customer(s) waiting, the server starts servicing the first customer in the queue. Otherwise it takes another vacation and so on. We study the steady state behaviour of the queue size distribution at random (stationary) point of time as well as at departure point of time and try to show that departure point queue size distribution can be decomposed into three independent random variables, one of which is the queue size of the standard M^x/M/1 queue. The interpretation of the other two random variables will also be provided. Further, we derive analytically explicit expressions for the system state (number of customers in the system) probabilities and provide their appropriate interpretations. Also, we derive some system performance measures. Finally, we develop a procedure to find mean waiting time of an arbitrary customer.     

On waiting for simultaneous access to two resources: Deterministic service distribution

Suppose that a test customer in anM/D/1queueing system can get service only if he has access to the server and a separate eventEhas occurred. All other customers only require access to the server. The time until the eventEoccurs is assumed to be an exponentially distributed random variable, if the test customer reaches the server beforeEoccurs, he must then return to the back of the queue. At any time, however, the test customer is allowed to give up his place in the queue and join the back of the queue. The test customer represents a computational task that depends upon the results of an associated task. The test customer's mean delay until service is derived assuming that he always maintains his position in the queue until he reaches the server. Conditions are given for which this "move-along" policy is optimal, i.e., minimizes the test customer's mean delay until service. A condition is also given for which the move-along policy is not optimal.     

Finite-source M/M/S retrial queue with search for balking and impatient customers from the orbit

The present paper deals with a generalization of the homogeneous multi-server finite-source retrial queue with search for customers in the orbit. The novelty of the investigation is the introduction of balking and impatience for requests who arrive at the service facility with a limited capacity and FIFO queue. Arriving customers may balk, i.e., they either join the queue or go to the orbit. Moreover, the requests are impatient and abandon the buffer after a random time and enter the orbit, too. In case of an empty buffer, each server searches for a customer in the orbit after finishing service. All random variables involved in the model construction are supposed to be exponentially distributed and independent of each other. The primary aim of this analysis is to show the effect of balking, impatience, and buffer size on the steady-state performance measures. Concentrating on the mean response time, several numerical examples are investigated by the help of the MOSEL-2 tool used for creating the model and calculating the stationary characteristics.     

Activity routing in a distributed supply chain: Performance evaluation with two inputs

In this paper, an industrial system is represented as a 2-input, three-stage queuing network. The two input queuing network receives orders from clients, and the orders are waiting to be served. Each order comprises (i) time of occurrence of the orders and (ii) quantity of items to be delivered in each order. The objective of this paper is to compute the minimum response time for the delivery of items to the final destination along the three stages of the network. The average number of items that can be delivered with this minimum response time constitute the optimum capacity of the queuing network. After getting serviced by the last node (a queue and its server) in each stage of the queuing network, a decision is made to route the items to the appropriate node in the next stage which can produce the least response time. Performance measures such as average queue lengths, average response times, and average waiting times of the jobs in the 2-input network are derived and plotted. Closed-form expressions for the equivalent service rate, equivalent average queue lengths, and equivalent response and waiting times of a single queue with a single server representing the 2-input queuing network are also derived and plotted.     

Queueing analysis and optimal control of and systems

We first consider a finite-buffer single server queue where arrivals occur according to batch Markovian arrival process (BMAP). The server serves customers in batches of maximum size ‘b’ with a minimum threshold size ‘a’. The service time of each batch follows general distribution independent of each other as well as the arrival process. We obtain queue length distributions at various epochs such as, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue length, mean waiting time, probability of blocking, etc. have been obtained. Total expected cost function per unit time is also derived to determine the optimal value N^* of N at a minimum cost for given values of a and b. Secondly, we consider a finite-buffer single server queue where arrivals occur according to BMAP and service process in this case follows a non-renewal one, namely, Markovian service process (MSP). Server serves customers according to general bulk service rule as described above. We derive queue length distributions and important performance measures as above. Such queueing systems find applications in the performance analysis of communication, manufacturing and transportation systems.     

Matrix analytical model of an ATM output buffer with self-similar traffic

Renewal processes with asymptotically hyperbolic interarrival time distributions are shown to exhibit self-similar behavior. An output buffer of an ATM switch is modeled as a discrete time queue with a single server, deterministic service times and self-similar renewal process input. A matrix geometric solution is found for the stationary distribution of states. For the case of hyperbolically distributed interarrival times, the mean and standard deviation of queue length are plotted for various values of the queue utilization and the self-similarity parameter of the arrival process. The self-similarity is found to have a significant impact on the performance of the queue.     

Discrete-time queues with generally distributed service times and renewal-type server interruptions

In this contribution, we investigate a discrete-time single-server queue subjected to server interruptions. Server interruptions are modeled as an on/off process with geometrically distributed on-periods and generally distributed off-periods. As message lengths can exceed one time-slot, different operation modes are considered, depending on whether service of an interrupted message continues, partially restarts or completely restarts after an interruption. For all alternatives, we establish expressions for the steady-state probability generating functions (pgf) of the buffer contents at message departure times and random slot boundaries, of the unfinished work at random slot boundaries, the message delay, and the lengths of the idle and busy periods. From these results, closed-form expressions for various performance measures, such as mean and variance of the buffer occupancy and message delay, can be established. As an application, we show that this model is able to assess performance of a multi-class priority scheduling system. We then illustrate our approach with some numerical examples.     

Steady state analysis of the GI/M/1/N queue with a variant of multiple working vacations

This paper studies the GI/M/1/N queue with a variant of multiple working vacations, where the server leaves for a working vacation as soon as the system becomes empty. The server takes at most H consecutive working vacations if the system remains empty after the end of a working vacation. Employing the supplementary variable and embedded Markov chain methods, we obtain the queue length distribution at different time epochs. Based on the various system length distribution, the probability of blocking, mean waiting times and mean system lengths have been derived. Finally, numerical results are discussed.     

Reducing congestion in bulk-service finite-buffer queueing system using batch-size-dependent service

In the design and analysis of any queueing system, one of the main objectives is to reduce congestion which can be achieved by controlling either arrival-rates or service-rates. This paper adopts the latter approach and analyzes a single-server finite-buffer queue where customers arrive according to the Poisson process and are served in batches of minimum size a with a maximum threshold limit b. The service times of the batches are arbitrarily distributed and depends on the size of the batches undergoing service. We obtain the joint distribution of the number of customers in the queue and the number with the server, and distributions of the number of customers in the queue, in the system, and the number with the server. Various performance measures such as the average number of customers in the queue (system) and with the server etc. are obtained. Several numerical results are presented in the form of tables and graphs and it is observed that batch-size-dependent service rule is more effective in reducing the congestion as compared to the one when service rates of the batches remain same irrespective of the size of the batch. This model has potential application in manufacturing, computer-communication network, telecommunication systems and group testing.     

Impatient customers in an M/M/1 queue with single and multiple working vacations

We consider an M/M/1 queue with impatient customers and two different types of working vacations. The working vacation policy is the one in which the server serves at a lower rate during a vacation period rather than completely stop serving. The customer’s impatience is due to its arrival during a working vacation period, in which the customer service rate is lower than the normal busy period. We analyze the queue for two different working vacation termination policies, a multiple working vacation policy and a single working vacation policy. Closed-form solutions and various performance measures like, the mean queue lengths and the mean waiting times are derived. The stochastic decomposition properties are verified for both multiple and single working vacation cases. A comparison of both the models is carried out to capture their performances with the change in system parameters.     

A repairable queueing model with two-phase service,start-up times and retrial customers

A repairable queueing model with a two-phase service in succession, provided by a single server, is investigated. Customers arrive in a single ordinary queue and after the completion of the first phase service, either proceed to the second phase or join a retrial box from where they retry, after a random amount of time and independently of the other customers in orbit, to find a position for service in the second phase. Moreover, the server is subject to breakdowns and repairs in both phases, while a start-up time is needed in order to start serving a retrial customer. When the server, upon a service or a repair completion finds no customers waiting to be served, he departs for a single vacation of an arbitrarily distributed length. The arrival process is assumed to be Poisson and all service and repair times are arbitrarily distributed. For such a system the stability conditions and steady state analysis are investigated. Numerical results are finally obtained and used to investigate system performance.