Modeling a supply chain using a network of queues

In this paper, a supply chain is represented as a two-input, three-stage queuing network. An input order to the supply chain is represented by two stochastic variables, one for the occurrence time and the other for the 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.     

Equivalent single-queue–single-server model for a Pentium processor

In this paper, a Pentium processor is represented as a queuing network. The objective of this paper is to deduce an equivalent single-queue–single-server model for the original queuing network. Closed-form expressions for the equivalent service rate, equivalent queue lengths, equivalent response and waiting times of the equivalent single-queue–single-server model are derived and plotted. For large values of arrival rate, queue lengths increase faster than the response times and waiting times for both the cases. Performance measures like, queue lengths, response times and waiting times are higher for lower service rates and lower for higher service rates (which is expected) of the different servers in the original queuing network. Also, the reliability in estimating performance measures for homogeneous workloads is much better than that for heterogeneous workloads.     

A closed queuing network model with single servers for multi-threaded architecture

In this paper, a closed queuing network model with single servers for each queue is proposed to model dataflow in a multi-threaded architecture. Multi-threading is useful in reducing the latency by switching among a set of threads in order to improve the processor utilization. Two sets of processors, synchronization and execution processors exist. Synchronization processors handle load/store operations and execution processors handle arithmetic/logic and control operations. A closed queuing network model is suitable for large number of job arrivals. The normalization constant is derived using a recursive algorithm for the given model. State diagrams are drawn from the closed queuing network model, and the steady-state balance equations are derived from it. Performance measures such as average response times and average system throughput are derived and plotted against the total number of processors in the closed queuing network model. Other important performance measures like processor utilizations, average queue lengths, average waiting times and relative utilizations are also derived.     

Performance evaluation of a single queue capable of handling two like jobs as a single entity

In this paper, we model an open queuing network and analyze performance measures with and without feedback as two individual cases. The model comprises a single queue with a dedicated processor capable of handling two like jobs as a single job. Two different job arrivals with different processing times are considered with an internal timer. Performance measures such as average queue length, average response time, average waiting time of the jobs are computed and plotted. The joint density function for the inter arrival time and arrival rate are derived. The probability mass function has been derived for all possible cases that may arise in a duration (0, t], considering n job arrivals during that period of time and an integer programming problem is formulated to obtain optimal sequence patterns which would maximize the efficiency of the model.     

Joint distributions in Poissonian tandem queues

A pair of single server queues arranged in series is considered. The input flow is Poisson and service times are mutually independent and exponentially distributed in each station. The joint distributions of the stationary waiting times and queue lengths at the arrival epoch are treated.     

On the M/G/1 queue with feedback and optional server vacations based on a single vacation policy

We study a single server queue with batch arrivals and general (arbitrary) service time distribution. The server provides service to customers, one by one, on a first come, first served basis. Just after completion of his service, a customer may leave the system or may opt to repeat his service, in which case this customer rejoins the queue. Further, just after completion of a customer's service the server may take a vacation of random length or may opt to continue staying in the system to serve the next customer. We obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers and the average waiting time in the queue. Some special cases of interest are discussed and some known results have been derived. A numerical illustration is provided.     

The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process

We consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process (BMAP). The server serves until system emptied and after that server takes a vacation. The server will take a maximum number H of vacations until either he finds at least one customer in the queue or the server has exhaustively taken all the vacations. We obtain queue length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue lengths and mean waiting times, etc. have been obtained. Several other vacation queueing models like, single and multiple vacation model, queues with exceptional first vacation time, etc. can be considered as special cases of our model.     

A Communication Multiplexer Problem: Two Alternating Queues with Dependent Randomly-Timed Gated Regime

Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace–Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling instants and at an arbitrary moment, mean busy period duration and mean cycle time length, expected number of messages transmitted during a busy period and mean waiting times. Finally, we present graphs of numerical results comparing the mean waiting times in the two queues as functions of the relative loads, showing the effect of the RTG regime.     

The serial correlation coefficients for waiting times in the stationary GI/M/m queue

Serial correlation coefficients are useful measures of the interdependence of successive waiting times. Potential applications include the development of linear predictors and determining simulation run lengths. This paper presents the algorithm for calculating such correlations in the multiserver exponential service queue, and relates it to known results for single server queues.     

Relating polling models with zero and nonzero switchover times

We consider a system ofN queues served by a single server in cyclic order. Each queue has its own distinct Poisson arrival stream and its own distinct general service-time distribution (asymmetric queues), and each queue has its own distinct distribution of switchover time (the time required for the server to travel from that queue to the next). We consider two versions of this classical polling model: In the first, which we refer to as the zero-switchover-times model, it is assumed that all switchover times are zero and the server stops traveling whenever the system becomes empty. In the second, which we refer to as the nonzero-switchover-times model, it is assumed that the sum of all switchover times in a cycle is nonzero and the server does not stop traveling when the system is empty. After providing a new analysis for the zero-switchover-times model, we obtain, for a host of service disciplines, transform results that completely characterize the relationship between the waiting times in these two, operationally-different, polling models. These results can be used to derive simple relations that express (all) waiting-time moments in the nonzero-switchover-times model in terms of those in the zero-switchover-times model. Our results, therefore, generalize corresponding results for the expected waiting times obtained recently by Fuhrmann [Queueing Systems 11 (1992) 109—120] and Cooper, Niu, and Srinivasan [to appear in Oper. Res.].Research supported in part by the National Science Foundation under grant DDM-9001751.     

离散时间服务台可修的排队系统MAP／PH（PH／PH）／1

本文研究离散时间可修排队系统,其中顾客的输入过程为离散马尔可夫到达过程（MAP）,服务台的寿命,服务台的顾客的服务时间和修理时间均为离散位相型（PH）变量,首先我们考虑广义服务过程,证明它是离散MAP,然后运用阵阵几何解理论,我们给出了系统的稳态队长分布和稳态等待时间分布,同时给出了系统的稳态可用度这一可靠性指标。     

Analysis of M/M/S queues with 1-limited service

In this paper, a multiple server queue, in which each server takes a vacation after serving one customer is studied. The arrival process is Poisson, service times are exponentially distributed and the duration of a vacation follows a phase distribution of order 2. Servers returning from vacation immediately take another vacation if no customers are waiting. A matrix geometric method is used to find the steady state joint probability of number of customers in the system and busy servers, and the mean and the second moment of number of customers and mean waiting time for this model. This queuing model can be used for the analysis of different kinds of communication networks, such as multi-slotted networks, multiple token rings, multiple server polling systems and mobile communication systems.     

Selecting batch size in discrete-time two-phase queuing system

This paper discusses the modeling and analysis of a discrete-time two-phase queuing system. Packets arrive at the system according to a Bernoulli process and receive batch service in the first queue and individual service in the second queue. We study the queue length, average waiting time of packets in the first queue and the effect of batch size on the waiting time.     

The polling system with a stopping server

This paper analyzes the polling system in which, unlike nearly all previous studies, the server comes to a stop when the system is empty rather than continuing to cycle. The possibility of server stopping permits a rich variety of alternatives for server behavior; we consider threestopping rules, governing server behavior when the system is emptied, and twostarting rules, governing server behavior when an arrival occurs to an idle system. The Laplace-Stieltjes Transforms and means for the waiting time andserver return time (the interval from an arrival at an unserved queue until the server returns to that queue) are determined. For the special case of zero changeover times and strictly cyclic service, explicit results are obtained.     

Analysis of Alternating-Priority Queueing Models with (Cross) Correlated Switchover Times

This paper analyzes a single server queueing system in which service is alternated between two queues and the server requires a (finite) switchover time to switch from one queue to the other. The distinction from classical results is that the sequence of switchover times from each of the queues need not be i.i.d. nor independent from each other; each sequence is merely required to form a stationary ergodic sequence. With the help of stochastic recursive equations explicit expressions are derived for a number of performance measures, most notably for the average delay of a customer and the average queue lengths under different service disciplines. With these expressions a comparison is made between the service disciplines and the influence of correlation is studied. Finally, through a number of examples it is shown that the correlation can significantly increase the mean delay and the average queue lengths indicating that the correlation between switchover times should not be ignored. This has important implications for communication systems in which a common communication channel is shared amongst various users and where the time between consecutive data transfers is correlated (for example in ad-hoc networks). In addition to this a number of notational mistakes in well-known existing literature are pointed out. AMS subject classification: 68M20, 60J85 A shorter version of this work has been published in the proceedings of IEEE Infocom 2005. This work was partly sponsored by the EURONGI network of excellence.     

Waiting times in a two-queue model with exhaustive and Bernoulli service

This paper deals with waiting times in a two-queue polling system in which one queue is served according to the Bernoulli service discipline and the other one attains exhaustive service. Exact results are derived for the LST's of the waiting time distributions via an iteration scheme. Based on those results the mean waiting times are expressed in the system parameters.     

The single server semi-markov queue

A general model for the single server semi-Markov queue is studied. Its solution is reduced to a matrix factorization problem. Given this factorization, results are obtained for the distributions of actual and virtual waiting times, queue lengths both at arrival epochs and in continuous time, the number of customers during a busy period, its length and the length of a busy cycle. Two examples are discussed for which explicit factorizations have been obtained.     

An approximation for multi-server queues with deterministic reneging times

This work was motivated by the timeout mechanism used in managing application servers in transaction processing environments. In such systems, a customer who stays in the queue longer than the timeout period is lost. We modeled a server node with a timeout threshold as a multi-server queue with Poisson arrivals, general service time distribution and deterministic reneging times. We proposed a scaling approach, and a fast and accurate approximation for the expected waiting time in the queue.     

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.