Tandem Queues Research Articles

Performance evaluation of WIP-controlled line production systems with constant processing times

We compared three types of WIP-controlled line production systems with constant processing times such as Kanban, CONWIP (constant work-in-process) and DBR (drum-buffer-rope). Based on the observation that such WIP-controlled line production systems are equivalent to m-node tandem queues with finite buffers under communication blocking policy, we applied a max-plus algebra based solution method for the tandem queue to evaluate their performance. Within our knowledge, this research is the first attempt to apply an exact solution method for comparing all three WIP-controlled line production systems at a time. Six-node numerical examples were also used to demonstrate the proposed analysis. The numerical results can be generalized and also provide some insights in designing production systems under certain limited condition.

Optimal Policies for a Dual-Sourcing Inventory Problem with Endogenous Stochastic Leadtimes

We consider a single-product, two-source inventory system with Poisson demand and backlogging. Inventory can be replenished through a normal supply source, which consists of a two-stage tandem queue with exponential production time at each stage. We can also place an emergency order by skipping the first stage, for a fee. There is no fixed order cost. There are linear order, holding and backorder costs. Through a new approach, we obtain optimal ordering policies for the discounted or long-run average cost, and also characterize near-optimal heuristic policies. The approach consists of four steps. The first step is to establish an equivalent system, in the sense that it has the same optimal policy as the original system. The second step is to construct a tandem queueing system, where costs are charged in accord with the equivalent system's cost structure. The third step derives an optimal control of the service rate at each server so as to minimize the tandem queue's system-wide cost. The fourth and final step is to translate the queue's optimal policy to an optimal policy for the equivalent system and hence the original system.

The Simulation and Study of the Operating System Efficiency Improvement of a Container Terminal

With the rapid expansion of Chinese container terminal throughput, each container terminal increasingly pays attention to the operational efficiency of storage yard. In this paper, aim at the problem of excessive containers relocation assignments to establish the slotting optimization allocation model, and verify the validity of the model by using Matlab; Using tandem queuing network modeling to build model of the container terminal operating system,and simulate the container terminal logistics system by Witness. It can provide a reference for efficiency improvement of the container terminal operating system.

Tollbooth tandem queues with infinite homogeneous servers

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

Tollbooth tandem queues with infinite homogeneous servers

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

Properties of Lagrangian distributions

Lagrange’s expansion is the power series expansion of the inverse function of an analytic function, and it leads to general Lagrangian distributions of the first kind as well as of the second kind. We present some theorems in which different sets of two analytic functions provide a Lagrangian distribution. Potential applicability of the theorem to tandem queueing process is studied.

Cyclic Queueing Networks With Subexponential Service Times and Finite Buffers

In this technical note, we consider closed tandem queueing networks with finite buffers and blocking. We assume that at least one station has subexponential service time distribution. We analyze this system under communication blocking and manufacturing blocking rules. Our objective is to derive expressions for the tail asymptotics of transient cycle times and waiting times. Furthermore, we study under which conditions on system parameters these tail asymptotics also hold for their stationary counter parts. Finally, we provide numerical examples to understand the convergence behavior of the tail asymptotics.

Simulation–optimization of complex tandem queue systems with reneging and server breakdowns considering budget constraints

This paper presents a simulation–optimization model for tandem G/G/K queuing systems with infinite capacity considering reneging and server breakdowns. In this problem, the entities that enter each queue after a certain amount of waiting time renege from the queue. Moreover, each of the servers may break down based on the failure rates and will be repaired immediately according to the repair rates. Owing to the complexity of these systems, the OptQuest approach that combines the scatter search (SS) and tabu search (TS) algorithms is employed to determine the optimal number of servers in each queue according to the capacity of resources as well as budget constraints. The main objective of this optimization is to minimize the total time in the system by keeping the average waiting time in each queue below the allowable waiting time. Finally, 20 distinct samples of tandem G/G/K queuing systems are used to evaluate the validity of the proposed simulation model and the optimization algorithm. To the best of our knowledge, this is the first study that both simulates and optimizes tandem G/G/K queuing systems by simultaneously considering reneging and server breakdowns such that budget constraints are satisfied.

Bounding tandem queuing system performance with variational theory

Queuing models are often used for traffic analysis, but analytical results concerning a system of queues are rare, thanks to the interdependence between queues. In this paper, we present an analysis of queuing systems to obtain bounds of their performance without studying the details of individual queues. Queuing dynamics is formulated in continuous-time, subject to variations of demands and bottleneck capacities. Our analysis develops new techniques built on the closed-form solution to a generalized queuing model for a single bottleneck. Taking advantage of its variational structure, we derive the upper and lower bounds for the total queue length in a tandem bottleneck system and discuss its implication for the kinematic wave counterpart. Numerical experiments are conducted to demonstrate the appropriateness of the derived upper and lower bounds as approximations in a stochastic setting.

Tandem system with retrials and impatient customers

Consideration was given to the tandem queuing system with multiple-line phases and a stationary Poisson flow. The customer arriving with the input flow and finding all first-phase servers occupied goes to an infinite-size orbit from which it retries to be serviced. Between the phases there is a finite buffer. The customers passing to the second phase have different priorities. For the highest-priority customers, second-phase servers are foreseen. If all places permitted for the given customer at the second phase are occupied, it either discharges the queuing system underserviced or returns to the first-phase orbit and retries to get a server at this phase. The impatient customers leaving the intermediate buffer behave in a similar manner. The stationary characteristics of system performance were established. Numerical examples, including the problem of optimal selection of the number of reserved servers, were presented.

ML and Bayes estimation in a two-phase tandem queue with a second optional service and random feedback

ABSTRACTIn this article, we consider a two-phase tandem queueing model with a second optional service and random feedback. The first phase of service is essential for all customers and after the completion of the first phase of service, any customer receives the second phase of service with probability α, feedback to the tail of the first queue with probability β if the service is not successful and leaves the system with probability 1 − α − β. In this model, our main purpose is to estimate the parameters of the model, traffic intensity, and mean system size, in the steady state, via maximum likelihood and Bayesian methods. Furthermore, we find asymptotic confidence intervals for mean system size. Finally, by a simulation study, we compute the confidence levels and mean length for asymptotic confidence intervals of mean system size with a nominal level 0.95.

Analysis of dual tandem queues with a finite buffer capacity and non-overlapping service times and subject to breakdowns

Tandem queues with a finite buffer capacity are the common structures embedded in practical production systems. We study the properties of tandem queues with a finite buffer capacity and non-overlapping service times subject to time-based preemptive breakdowns. Different from prior aggregation and decomposition approaches, we view a tandem queue as an integrated system and develop an innovative approach to analyze the performance of a dual tandem queue through the insight from Friedman's reduction method. We show that the system capacity of a dual tandem queue with a finite buffer and breakdowns can be less than its bottleneck-sees-initial-arrivals system due to the existence of virtual interruptions. Furthermore, the virtual interruptions depend on job arrival rates in general. Approximate models are derived using priority queues and the concept of virtual interruptions.

An Approximation Method for Blocking Probabilities in M/D/1/K1→ ⋅/D/1/K2Queues

Obtaining exact blocking probabilities for tandem queues with finite capacities is not a trivial problem. In this paper, we propose a computational approximation method using max-plus algebra for computing blocking probability in a Poisson-driven 2-node tandem queue with finite capacities and constant service times. The blocking probability of a finite-capacity queueing system can be obtained from either the tail probability of stationary waiting time or the difference between two expected stationary waiting times at the first node of the corresponding extended 3-node tandem queue. The computational results in this study show that the proposed approach provides a good approximation of the blocking probability, and in particular, it works well under moderately to heavily loaded situations. The proposed approach is not limited to a particular blocking policy, system structure, or service time; hence, it is applicable to general queues with finite buffer capacities and various blocking policies.

Dependence among single stations in series and its applications in productivity improvement

Theory of constraints has been commonly used in production systems to improve productivity. Since the improvement on an upstream workstation may have impact on its downstream servers, finding the true bottleneck is not trivial in a stochastic production line. Due to the analytical intractability of general tandem queues, we develop methods to quantify the dependence among stations through simulation. Dependence is defined by the contribution queue time at each station, and contribution factors are developed based on the insight from Friedman's reduction method and Jackson networks. In a tandem queue, the dependence among stations can be either diffusion or blocking, and their impact depends on the positions relative to the bottlenecks. Based on these results, we show that improving the performance of the system bottleneck may not be the most effective place to reduce system cycle time. Rather than making independence assumptions, the proposed method points out a promising direction and sheds light on the insights of the dependence in practical systems.

A closed-form solution for a tollbooth tandem queue with two heterogeneous servers and exponential service times

This paper considers a tollbooth queueing system where customers arrive according to a Poisson process and there are two heterogeneous servers of exponential service times. We show that eigenvalues can be found explicitly for the characteristic matrix polynomial associated with the Markov chain characterizing the system. We derive a closed-form solution for the steady state probabilities to make the straightforward computation of performance measures.

Minimizing the Delay Time of Data Transfer in Access Networks Using Queue Models

Modeling and performance evaluation are becoming critical issues in access networks. The upstream channel of cable access network regulated by request grant mechanism can viewed as a two stage tandem queue. The intention of this paper is to divide the capacity of the upstream data among the two stations to ensure minimum delay time by considering the first node as M/M/1 queue and second node as M/M/1 retrial queue.

An analysis of nonstationary coupled queues

We consider a two dimensional time varying tandem queue with coupled processors. We assume that jobs arrive to the first station as a non-homogeneous Poisson process. When each queue is non-empty, jobs are processed separately like an ordinary tandem queue. However, if one of the processors is empty, then the total service capacity is given to the other processor. This problem has been analyzed in the constant rate case by leveraging Riemann Hilbert theory and two dimensional generating functions. Since we are considering time varying arrival rates, generating functions cannot be used as easily. Thus, we choose to exploit the functional Kolmogorov forward equations (FKFE) for the two dimensional queueing process. In order to leverage the FKFE, it is necessary to approximate the queueing distribution in order to compute the relevant expectations and covariance terms. To this end, we expand our two dimensional Markovian queueing process in terms of a two dimensional polynomial chaos expansion using the Hermite polynomials as basis elements. Truncating the polynomial chaos expansion at a finite order induces an approximate distribution that is close to the original stochastic process. Using this truncated expansion as a surrogate distribution, we can accurately estimate probabilistic quantities of the two dimensional queueing process such as the mean, variance, and probability that each queue is empty.

Appointment scheduling in tandem-type service systems

Appointment-based service systems arise in a broad variety of healthcare settings (for example an outpatient clinic or a dentist). Where most existing algorithms specifically consider the situation of the patient undergoing a single service, in many practical situations multiple services have to be sequentially performed. Modeling the service system as a tandem queue, the main objective of this paper is to generate schedules that soundly balance the interests of patients (i.e., low waiting times) and staff (i.e., low idle times). Importantly, following up on prior work for the single-node queue, we advocate a phase-type based technique that can deal with any service-time distribution (which may, in addition, vary across patients). Relying on a novel recursive scheme to evaluate the sojourn-time distribution of clients in such tandem systems, we show how optimal schedules can be computed. Our technique is illustrated by extensive numerical experimentation, also leading to practical guidelines that apply to a broad range of parameter settings.

Explicit expressions for moments of waiting times in Poisson driven deterministic two-node tandem queues with blocking

There has been no analytic expression for a multi-node queue even with constant processing times because of a correlation caused by blocking between nodes. This study introduces explicit expressions for moments and tail probability of stationary waiting times in a Poisson-driven deterministic 2-node tandem queue with blocking. Unlike the normal queueing theory, we derive these expressions from the previous results based on the max-plus algebraic approach. Two blocking policies are considered: blocking before service (BBS) and blocking after service (BAS).

Dynamic Server Assignment With Task-Dependent Server Synergy

We study tandem queueing systems with finite buffers in which servers work more efficiently in teams than on their own and the synergy among collaborating servers can be task-dependent. Our goal is to determine the dynamic server assignment policy that maximizes the long-run average throughput. When each server works with the same ability at each task that she/he is assigned to, we show that any nonidling policy where all servers work in teams of two or more at all times is optimal. On the other hand, when the server abilities are task-dependent, we show that for Markovian systems with two stations and two servers, depending on the synergy among the servers, the optimal policy either assigns the two servers to different stations when possible, or lets them work in a team at all times. Finally, for larger Markovian systems, we provide sufficient conditions that guarantee that the optimal policy has all servers working together at all times.