Tandem Queueing System Research Articles

Tandem queueing system with infinite and finite intermediate buffers and generalized phase-type service time distribution

A tandem queueing system with infinite and finite intermediate buffers, heterogeneous customers and generalized phase-type service time distribution at the second stage is investigated. The first stage of the tandem has a finite number of servers without buffer. The second stage consists of an infinite and a finite buffers and a finite number of servers. The arrival flow of customers is described by a Marked Markovian arrival process. Type 1 customers arrive to the first stage while type 2 customers arrive to the second stage directly. The service time at the first stage has an exponential distribution. The service times of type 1 and type 2 customers at the second stage have a phase-type distribution with different parameters. During a waiting period in the intermediate buffer, type 1 customers can be impatient and leave the system. The ergodicity condition and the steady-state distribution of the system states are analyzed. Some key performance measures are calculated. The Laplace–Stieltjes transform of the sojourn time distribution of type 2 customers is derived. Numerical examples are presented.

A dual tandem queueing system with GI service time at the first queue

In this paper we consider the analysis of a tandem queueing model$M/G/1 -> ./M/1$. In contrast to the vast majority of the previousliterature on tandem queuing models we consider the case with GIservice time at the first queue and with infinite buffers. Thesystem can be described by an M/G/1-type Markov process at thedeparture epochs of the first queue. The main result of the paper isthe steady-state vector generating function at the embedded epochs,which characterizes the joint distribution of the number ofcustomers at both queues. The steady-state Laplace-Stieljestransform and the mean of the sojourn time of the customers in thesystem are also obtained.  &nbsp We provide numerical examples and discuss the dependency of thesteady-state mean of the sojourn time of the customers on severalbasic system parameters. Utilizing the structural characteristics ofthe model we discuss the interpretation of the results. This givesan insight into the behavior of this tandem queuing model and can bea base for developing approximations for it.

A tandem [formula omitted] queue with blocking

Tandem queues are widely used in mathematical modeling of random processes describing the operation of manufacturing systems, supply chains, computer and telecommunication networks. Although there exists a lot of publications on tandem queueing systems, analytical research on tandem queues with non-Markovian input is very limited. In this paper, the results of analytical investigation of two-node tandem queue with arbitrary distribution of inter-arrival times are presented. The first station of the tandem is represented by a single-server queue with infinite waiting room. After service at the first station, a customer proceeds to the second station that is described by a single-server queue without a buffer. Service times of a customer at the first and the second server have PH (Phase-type) distributions. A customer, who completes service at the first server and meets a busy second server, is forced to wait at the first server until the second server becomes available. During the waiting period, the first server becomes blocked, i.e., not available for service of customers. We calculate the joint stationary distribution of the system states at the embedded epochs and at arbitrary time. The Laplace–Stieltjes transform of the sojourn time distribution is derived. Key performance measures are calculated and numerical results presented.

Tandem queueing system with impatient customers as a model of call center with Interactive Voice Response

A tandem queueing system with a Markovian Arrival Process (MAP) useful in modeling a call center with Interactive Voice Response (IVR) is investigated. The first stage has a finite number of servers without buffer while the second stage of the tandem has a finite buffer and a finite number of servers. The service time at the first (second) stage has an exponential (phase type) distribution. A special approach for reducing the number of states of the stochastic process that describes the behavior of the system is used. The main performance measures are calculated. The Laplace–Stieltjes transform of the sojourn time distribution is derived. The numerical results are presented.

Analysis of a tandem queue with heterogeneous servers subject to catastrophes

This study analyzed a tandem queueing system with blocking and no waiting line. The arrival process is assumed to be Poisson with rate. There are two non-identical servers in the system. The service times are exponentially distributed with parameters and at servers 1 and 2, respectively. Besides, the catastrophes occur in a Poisson manner with rate in the system. When server 1 is busy or blocked, the customer who arrives in the system leaves the system without being served. Such customers are called “lost customers”. The probability of losing a customer was computed for the system. Key words: Tandem queueing, heterogeneous servers, catastrophes, loss probability.

A hybrid approach to minimize state space explosion problem for the solution of two stage tandem queues

Two stage open queuing networks are used for modeling the subsystem-behaviour in computers and communication networks, mass storage devices, memory servers, and queuing analysis of wireless mobile cellular networks. The queuing analysis of wireless systems is essential in order to quantify the impact of different factors on quality of service (QoS); performance measures so that wireless protocols can be designed and/or tuned in an optimal manner. In that sense two stage open queuing systems are particularly important to model handoff phenomena, especially for the integration of two different systems such as cellular and wireless local area networks (WLANs). Analytical solutions for two-dimensional Markov processes suffer from the state space explosion problem. The numerical difficulties caused by large state spaces, make it difficult to handle multiple servers at the second stage of a tandem queuing system together with server failures and repairs. This study presents a new approach to analytical modeling of open networks offering improvements in alleviating this problem. The proposed solution is a hybrid version, which combines well known spectral expansion, and hierarchical Markov reward rate approaches. Using this approach, two-stage open networks with multiple servers, break-downs, and repairs at the second stage and feedback can be modeled as three-dimensional Markov processes and solved for performability measures. Comparative results show that the new algorithm used for solution, provides a high degree of accuracy, and it is computationally more efficient than the existing approaches. The proposed model is capable of solving other three-dimensional Markov processes.

An explicit solution for a tandem queue with retrials and losses

We consider a retrial tandem queueing system with two servers whose service times follow two exponential distributions. There are two types of customers: type one and type two. Customers of type one arrive at the first server according to a Poisson process. An arriving customer of type one that finds the first server busy joins an orbit and retries to enter the server after some time. We assume that the arrival rate of customers from the orbit is a linear function of the number of retrial customers. After being served at the first server, a customer of type one moves to the second server. Customers of type two directly arrive at the second server according to another Poisson process. Customers of both types one and two are lost if the second server is busy upon arrival. For this model, we derive explicit expressions of the joint stationary distribution between the number of customers in the orbit and the states of the servers. We prove that the stationary distribution is computed by a numerically stable algorithm. Numerical examples are provided to show the influence of parameters on the performance of the system.

On the admission control and demand management in a two-station tandem production system

This paper considers a two-station tandem production system consisting of make-to-stock and make-to-order facilities. The make-to-stock facility produces componentswhich are served for external demands as well as internal make-to-order operations while the make-to-order facility processes customer orders with the option toaccept or reject. We address the problem of coordinating the decision of when to accept customer order and when to satisfy component demand that maximizes the totalexpected discounted profit. To deal with this issue, we present a Markov decision process model of two-station tandem queueing system and characterize the structureof the optimal policy. We investigate the marginal impacts of system parameters on the optimal policy and implement a numerical experiment for comparing theperformance between the optimal policy and the static policy with two fixed thresholds.

Tandem service system with batch Markov flow and repeated calls

Consideration was given to the tandem queuing system (QS) whose first stage consists of a single-server system with batch Markov arrivals and repeated calls and the second stage consists of a multi-server QS without buffer. In the case of no idle servers in the second stage, the customer serviced at the first stage either waits for server release and at that blocks the first-stage service or discharges the system being serviced incompletely. The stationary distribution of the probabilities of system states at the embedded and arbitrary time instants and the main characteristics of system performance were established. Numerical examples were presented.

Dynamic production control in a serial line with process queue time constraint

This research examines the production control problem in two-station tandem queueing systems under time constraints. In these two-station tandem queueing systems, jobs must first be processed at the upstream station and then the downstream station. For each job, the sum of the waiting and processing time in the downstream queue is limited by an upper bound. This time constraint is called the process queue time constraint. When the process queue time constraint is violated, a significant scrap cost will be accrued. In this research, we develop a Markov decision model to study the production control problem under process queue time constraints. The objective is to minimise the sum of the expected long-run average inventory holding costs and scrap costs. According to the Markov decision model, an interesting exhaustive structure of the optimal production control policy is found. Based on this exhaustive structure, an efficient algorithm is developed to solve the production control problem numerically. The performance of the proposed algorithm is verified by a simulation study. Compared with other heuristics in the literature, the proposed algorithm can significantly reduce production costs while improving system throughput and utilisation.

Allocating flexible servers in serial systems with switching costs

Consider a firm that operates a make-to-order serial production system and employs a cross-trained workforce. We model such a firm as a tandem queuing system in which flexible servers can be allocated across stations, and assume that a switching cost is charged when servers move between stations. We show that even in the two-station two-server case the optimal policy follows a complex state-dependent structure that may be difficult to implement in practice. We propose three alternate heuristic policies and assess their performance. We show that a simpler policy which only moves one server can achieve close to optimal results.

A tandem retrial queueing system with two Markovian flows and reservation of channels

We consider a tandem queueing system with single-server first station and multi-server second station. The input flow at Station 1 is described by the BMAP (batch Markovian arrival process). Customers from this flow are considered as non-priority customers. Customers of an arriving group, which meet a busy server, go to the orbit of infinite size. From the orbit, they try their luck in exponentially distributed random time. Service times at Station 1 are independent identically distributed random variables having an arbitrary distribution. After service at Station 1 a non-priority customer proceeds to Station 2. The service time by a server of Station 2 is exponentially distributed. Besides customers proceeding from Station 1, an additional MAP flow of priority customers arrives at Station 2 directly, not entering Station 1. If a priority customer meets a free server upon arrival, it starts service immediately. Else, it leaves the system forever. It is assumed that a few servers of Station 2 are reserved to serve the priority customers only. We calculate the stationary distribution and the main performance measures of the system. The problem of optimal design is numerically investigated.

Analysis of a two-node task-splitting feedback tandem queue with infinite buffers by functional equation

In this article, a two-node single-processor Markovian tandem queueing system with task splitting and feedback is considered. Each node has an infinite buffer before it and, thus, no blocking is possible in the system. Splitting feature is added to the model considered and it makes it a novel tandem queue. The functional equation developed from the generating function applied to the system of difference equations is solved using Riemann-Hilbert problem. The mean of the stationary queue length at each node is found. An Algorithm is given for computation of performance measures. Using approximation, a numerical example is offered to illustrate the workability of the algorithm.

A tandem queueing system with applications to pricing strategy

In this paper, we analyze a Markovian queueing system with multipletypes of customers and two queues in tandem.All customers have to go through two stages of services.In Stage 1, the queueing system has multiple identical servers while in Stage 2,there is one single-server queue for each type of customers.The queueing discipline in the whole system is Blocked Customer Delayed (BCD).We first obtain the steady-state probability distribution of thequeueing system and the expected waiting time for customers.We then apply the queueing model to solve an optimal pricing policy problemin assuming that the demand rate is dependent on the price.The objective is to minimize the number of serversin the first stage and also maximize the expected earningsby taking into account the demand and the prices.We also obtained some analytic results for the optimal pricing strategy.

Optimal stochastic scheduling of two interconnected queues with varying service rates

We consider two-stage tandem queueing systems attended by two specialized and one flexible server, where all servers have time varying rates. Assuming exponential processing times and linear holding costs, we derive properties of server allocation policies that minimize expected costs over an infinite time horizon.

A tandem queueing system with task-splitting, feedback, and blocking

A novel two-station single-processing tandem queueing system with task splitting, feedback, and blocking is considered. Two cases are analysed: a finite buffer at each station, and only the intermediate buffer finite. After completing processing at each station, a task may leave the system, join the other station, go for split, go to the other station or return to its own station. The steady-state joint distribution of the queue lengths that will lead to the moments of the queue lengths, mean waiting time, probabilities of each station and the system being busy and being idle are obtained through algorithms. A numerical example is provided.

Dynamic Allocation of Reconfigurable Resources in a Two-Stage Tandem Queueing System With Reliability Considerations

Consider a two-stage tandem queueing system, with dedicated machines in each stage. Additional reconfigurable resources can be assigned to one of these two stations without setup cost and time. In a clearing system (without external arrivals) both with and without machine failures, we show the existence of an optimal monotone policy. Moreover, when all of the machines are reliable, the switching curve defined by this policy has slope greater than or equal to -1. This continues to hold true when the holding cost rate is higher at the first stage and machine failures are considered.

On the Introduction of an Agile, Temporary Workforce into a Tandem Queueing System

We consider a two-station tandem queueing system where customers arrive according to a Poisson process and must receive service at both stations before leaving the system. Neither queue is equipped with dedicated servers. Instead, we consider three scenarios for the fluctuations of workforce level. In the first, a decision-maker can increase and decrease the capacity as is deemed appropriate; the unrestricted case. In the other two cases, workers arrive randomly and can be rejected or allocated to either station. In one case the number of workers can then be reduced (the controlled capacity reduction case). In the other they leave randomly (the uncontrolled capacity reduction case). All servers are capable of working collaboratively on a single job and can work at either station as long as they remain in the system. We show in each scenario that all workers should be allocated to one queue or the other (never split between queues) and that they should serve exhaustively at one of the queues depending on the direction of an inequality. This extends previous studies on flexible systems to the case where the capacity varies over time. We then show in the unrestricted case that the optimal number of workers to have in the system is non-decreasing in the number of customers in either queue.

A MATHEMATICAL PROGRAMMING APPROACH TO FUZZY TANDEM QUEUES

Tandem queueing models play an important role in many real world systems such as computer systems, production lines, and service systems. This paper proposes a procedure to construct the membership functions of the performance measures in tandem queueing systems, in that the arrival rate and service rates are fuzzy numbers. The basic idea is to transform a fuzzy tandem queue to a family of crisp tandem queues by applying the α -cut approach. Then on the basis of α -cut representation and the extension principle, a pair of mathematical programs is formulated to describe this family of crisp tandem queues, via which the membership functions of the performance measures are derived. Two numerical examples are solved successfully to demonstrate the validity of the proposed approach. Since the performance measures are expressed by membership functions rather than by crisp values, the fuzziness of input information is completely conserved. Thus the proposed approach for fuzzy systems can represent the system more accurately, and more information is provided for designing queueing systems. The successful extension of tandem queues to fuzzy environments permits tandem queueing models to have wider applications.

Performance modelling of GPRS with bursty multiclass traffic

An analytic framework is devised, based on the principle of maximum entropy (ME), for the performance modelling and evaluation of a wireless GSM/GPRS cell supporting bursty multiple class traffic of voice calls and data packets under complete partitioning (CPS), partial sharing (PSS) and aggregate sharing (ASS) traffic handling schemes. Three distinct open queueing network models (QNMS) under CPS, PSS and ASS, respectively, are described, subject to external compound Poisson traffic processes and generalised exponential (GE) transmission times under a repetitive service blocking mechanism and a complete buffer sharing management rule. Each QNM generally consists of three building block stations, namely a loss system with GSM/GPRS traffic and a system of access and transfer finite capacity queues in tandem dealing with GPRS traffic under head-of-line and discriminatory processor sharing scheduling disciplines, respectively. The analytic methodology is illustrated by focusing on the performance study of the GE-type tandem queueing system for GPRS under a CPS. An ME product-form approximation is characterised leading into a decomposition of the tandem system into individual queues and closed-form ME expressions for state and blocking probabilities are presented. Typical numerical examples are included to validate the ME solutions against simulation and study the effect of external GPRS bursty traffic upon the performance of the cell. Moreover, an overview of recent extensions of the work towards the analysis of a GE-type multiple server finite capacity queue with preemptive resume priorities and its implications towards the performance modelling and evaluation of GSM/GPRS cells with PSS and ASS are included.