On multiple priority multi-server queues with impatience

This paper studies a queueing system with a Markov arrival process with marked arrivals and PH-distribution service times for each type of customer. Customers (regardless of their types) are served on a mixed first-come-first-served (FCFS) and last-come-first-served (LCFS) nonpreemptive basis. That is, when the queue length is N (a positive integer) or less, customers are served on an FCFS basis; otherwise, customers are served on an LCFS basis. The focus is on the stationary distribution of queue strings, busy periods, and waiting times of individual types of customers. A computational approach is developed for computing the stationary distribution of queue strings, the mean of busy period, and the means and variances of waiting times. The relationship between these performance measures and the threshold number N is analyzed in depth numerically. It is found that the variance of the virtual (actual) waiting time of an arbitrary customer can be reduced by increasing N . c © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 399–421, 2000

We establish a conservation law for G/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in a G/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in a GI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

A novel flexible discipline for providing priority in a single-server queue is applied to the system where the service can be unreliable what results in the loss of a customer or repetition of its service. According to this discipline, arriving customers are stored in the finite buffers dedicated to the customers of the corresponding type if the buffer is not full. After the staying in the corresponding buffer during the exponentially distributed time, the customers try to enter the main buffer of a finite capacity which is common for both types of customers. If this main buffer is full, the customer returns to the dedicated buffer and repeats the attempts to enter the main buffer later. Customers staying in the dedicated buffers are impatient and can go away from the system after a certain patience period duration of which has an exponential distribution. Customers of both types, which succeed to enter the main buffer, are picked up for the service in the order of their admission to this buffer. Providing the preference to priority customers is managed via the corresponding choice of capacities of the dedicated buffers and the rate of trials to transfer to the main buffer. Performance measures of this system are obtained under the assumption that the arrival flows of two types of customers are defined by the Markov arrival processes and the service time has the distribution of phase-type with failures type. Some aspects relating to an optimal choice of the parameters of the system are discussed via numerical experiments.

We consider an infinite sequence of customers of typesand an infinite sequence of servers of typeswhere a server of typejcan serve a subset of customer typesC(j) and where a customer of typeican be served by a subset of server typesS(i). We assume that the types of customers and servers in the infinite sequences are random, independent, and identically distributed, and that customers and servers are matched according to their order in the sequence, on a first-come–first-served (FCFS) basis. We investigate this process of infinite bipartite matching. In particular, we are interested in the rateri,jthat customers of typeiare assigned to servers of typej. We present a countable state Markov chain to describe this process, and for some previously unsolved instances, we prove ergodicity and existence of limiting rates, and calculateri,j.

Performance analysis of broadcast star network with collision-avoidance switch

This paper considers a continuous-time queueing model with two types (classes) of customers each having their own dedicated server. The objective is to have a better grasp on the concept of a global first-come-first-served service discipline with presorting, i.e., all arriving customers are accommodated in one single FCFS queue, regardless of their type, with an exception of the first P customers. For the first P customers the FCFS rule holds only within the type, i.e., customers of different types can overtake each other in order to be served. Due to the global FCFS rule the model becomes non-workconserving and on the other hand we also have to keep track of the types of customers in the first P customers. The motivation of our work is the concept of a turn lane in road traffic, i.e., a lane reserved for vehicles making a specific turn at the next junction. This paper intends to be a step towards an analytic model to aid in the decision process of various policy makers of the optimal length of turn lanes.

In this paper, we study the total workload process and waiting times in a queueing system with multiple types of customers and a first-come-first-served service discipline. An M/G/1 type Markov chain, which is closely related to the total workload in the queueing system, is constructed. A method is developed for computing the steady state distribution of that Markov chain. Using that steady state distribution, the distributions of total workload, batch waiting times, and waiting times of individual types of customers are obtained. Compared to the GI/M/1 and QBD approaches for waiting times and sojourn times in discrete time queues, the dimension of the matrix blocks involved in the M/G/1 approach can be significantly smaller.

The subject of this paper is a two-station mixed queueing network with two customer types: “Open” customers enter the network at station 1 and depart the system after receiving service. Meanwhile, a fixed number of “closed” customers circulate between stations 1 and 2. Such a mixed queueing network model can represent a single-stage production system that services both make-to-order and make-to-stock customers. We present fluid and diffusion limits for this network under the first-in-first-out service discipline. We find that the heavy traffic limit of the workload process at station 1 is a reflected Brownian motion (RBM) on a finite interval. This result is in contrast to the behavior of the original mixed network model, in which the workload at station 1 need not be bounded.

A multiserver queueing system with infinite and finite buffers, two types of customers, and two types of servers as a model of a call center with a call-back for lost customers is investigated. Type 1 customers arrive to the system according to a Markovian arrival process. All rejected type 1 customers become type 2 customers. Typer,r=1,2, servers serve typercustomers if there are any in the system and serve typer′,r′=1,2, r′≠r,customers if there are no typercustomers in the system. The service times of different types of customers have an exponential distribution with different parameters. The steady-state distribution of the system is analyzed. Some key performance measures are calculated. The Laplace-Stieltjes transform of the sojourn time distribution of type 2 customers is derived. The problem of optimal choice of the number of each type servers is solved numerically.

In this paper, we analyze a multi-server queueing system with a marked Markov arrival process of two types of customers and a phase-type distribution of service time depending on the type of customer. Customers of both types are assumed to be impatient and renege from the buffers after an exponentially distributed number of times. The strategy of flexible provisioning of priorities is analyzed. It assumes a randomized choice of the customers from the buffers, with probabilities dependent on the relation between the number of customers in a priority finite buffer and the fixed threshold value. To simplify the construction of the underlying Markov chain and the derivation of the explicit form of its generator, we use the so-called generalized phase-type distribution. It is shown that the created Markov chain fits the category of asymptotically quasi-Toeplitz Markov chains. Using this fact, we show that the considered Markov chain is ergodic for any value of the system parameters and compute its stationary distribution. Expressions for key performance measures are presented. Numerical results that show how the parameters of the control strategy affect the system’s performance measurements are given. It is shown that the results can be used for managerial purposes and that it is crucial to take correlation in the arrival process into account.

A single-server queueing system with a marked Markovian arrival process of heterogeneous customers is considered. Type-1 customers have limited preemptive priority over type-2 customers. There is an infinite buffer for type-2 customers and no buffer for type-1 customers. There is also a finite buffer (stock) for consumable additional items (semi-products, half-stocks, etc.) which arrive according to the Markovian arrival process. Service of a customer requires a fixed number of consumable additional items depending on the type of the customer. The service time has a phase-type distribution depending on the type of the customer. Customers in the buffer are impatient and may leave the system without service after an exponentially distributed amount of waiting time. Aiming to minimize the loss probability of type-1 customers and maximize throughput of the system, a threshold strategy of admission to service of type-2 customers is offered. Service of type-2 customer can start only if the server is idle and the number of consumable additional items in the stock exceeds the fixed threshold. Stationary distributions of the system states and the waiting time are computed. In the numerical example, we show some interesting effects and illustrate a possibility of application of the presented results for solution of optimization problems.

This paper deals with the Markovian queueing system with working vacation in which the server may encounter with breakdowns. The repair process does start immediately after a breakdown. During this period any new customer is allowed to join the system. Whenever the server finds nobody, the server goes for a working vacation during which the server renders service at a slower rate than the normal. Further, we considered two types of customer's impatience Balking and Reneging. We solved the system of differential equations to find Transient state probabilities and computed various performance indices like expected length of system, the mean waiting time etc. We then performed the sensitivity analysis and observed the impact on different parameters.

Implementable algorithms are developed for a single server queue serving two types of customers and operating under a dynamic non-preemptive priority rule which assigns higher priority to type 1 customers when their queue length exceeds a pre-assigned theshold N, and to type 2 customers otherwise. The model is discussed under the assumption of a finite waiting room of size K-1 for type 2 customers and for the case of general service times and Poisson arrivals. The recursive computational schemes, which are obtained here using the function-analytic methods recently introduced by M. F. Neuts, can be used interactively in the design of such systems for arriving at desirable values of N and K. A self-contained section of numerical examples illustrates some interesting qualitative aspects of such a model.

Bulk arrival retrial G-queue with impatient customers and multi-services subject to server breakdowns has been analyzed. The system allows the arrival of two types of customers: positive customers and negative customers in the system. The negative customers make the server fail if they find the server in busy state, whereas positive customers are served if the server is idle otherwise they join the virtual pool of customers called orbit. The customers from the retrial orbit try their chance again for the service. The customers have the option of obtaining more than one service. Moreover, the customers are impatient and may renege from the system with probability \((1-r)\). The server is sent for repair as soon as it breakdowns; after repair, the service process starts again. Also, the server has the provision to initiate the service when there are N customers accumulated in the system. Using supplementary variables technique and generating functions, various performance measures like reliability indices and long run probabilities have been obtained.

This paper deals with the Markovian queueing system with working vacation in which the server may met with failures. There will be no delay in initiating repair. During this period any new customer is allowed to join the system. Each customer will be provided Erlangian phase service. Whenever the server finds nobody, the server goes for a working vacation during which the server provides service at a slower rate than the normal. Further, we considered two types of customer’s impatience. We solved the system of differential equations to find Transient state probabilities and computed various performance indices of the queue. We then executed the sensitivity analysis and observed the impact on different parameters.