Tandem Queueing System Research Articles

Optimal Pricing for Tandem Queues with Finite Buffers

We consider optimal pricing for a two-station tandem queueing system with finite buffers and price-sensitive customers. The service provider quotes prices to customers using either static or dynamic pricing. The objective is to maximize either the infinite-horizon discounted profit or the long-run average profit of the provider. We show that there exists an optimal dynamic policy that exhibits an interesting monotone structure, in which the quoted prices have greater dependency on the queue length at station 1 than at station 2, for both the discounted and long-run average problems. We then study the long-run average problem and show that the optimal static policy performs as well as the optimal dynamic policy when the buffer size at station 1 becomes large, there are no holding costs, and the arrival rate is either small or large. We learn from numerical results that for systems with small arrival rates, the optimal static policy is near optimal even when the buffer at station 1 is small and there are positive holding costs. On the other hand, for systems with arrival rates that are not small, there are cases where the optimal dynamic policy performs much better than the optimal static policy.

Optimal Dynamic Allocation of Collaborative Servers in Two Station Tandem Systems

We consider two-stage tandem queueing systems with one dedicated server in each station and a flexible server that can serve both stations. We assume exponential service times, linear holding costs accrued by jobs present in the system, and a collaborative work discipline. We seek optimal server allocation strategies for systems without external arrivals (clearing systems). When the combined rate of collaborating servers is less than the sum of their individual rates (partial collaboration), we identify conditions under which the optimal server allocation strategy is non-idling and has a threshold-type structure. Our results extend previous work on systems with additive service rates. When the aforementioned conditions are not satisfied we show by examples that the optimal policy may have counterintuitive properties, which is not the case when a fully collaborative service discipline is assumed. We also obtain novel results for any type of collaboration when idling policies may be optimal.

Optimization of admission control in tandem queue with heterogeneous customers and pre-service

ABSTRACTA new type of tandem queueing systems with heterogeneous customers is analysed. Tandem consists of two stations each of which is modelled by a multi-server queueing system. Type-1 customers need service at station 1. After service at this station, a type-1 customer spends a random amount of time in the so-called pre-service area and then moves for service at station 2. Type-2 customers need service only at station 2 and have lower priority. A strategy of their admission depending on the current number of customers in the pre-service area and station 2 is offered. This strategy is defined by three integers (thresholds). Cost criterion, which defines the quality of the system operation, is the weighted sum of the mean numbers of lost customers from the pre-service area and type-2 customers per unit of time. The most difficult part of solving the problem of minimizing the cost criterion consists of computation of the stationary distribution of the system states under the fixed set of the thresholds. Numerical results are presented. Results can be used for optimization of various real-world systems such as manufacturing, transportation and health-care systems as well as for optimization of booking systems in HORECA (hotel/restaurant/cafe), museums, etc.

Optimization of Two-Phase Queuing System and Its Application to the Control of Data Transmission between Two Robotic Agents

A tandem queuing system that contains two single-channel stations with finite buffers and allows blocking of the first server is considered. The first station receives nonstationary Poisson packet flow that is processed at a controlled rate. In the case of the queue overflow in the first system, the input packet is lost. The second station does not allow overflow due to control of the acceptance probability (a decrease in such a probability leads to slowing of packet sending from the first station). The queuing system is described with the aid of the controlled Markov process. The optimal control problem is considered over a finite time horizon using the criterion of minimum average losses under the constraints on the total service time and energy consumption of the first station. Optimization algorithms are proposed for synthesis of the control law for nonstationary data flow in two-agent robotic system.

Analysis of the Optimal Resource Allocation for a Tandem Queueing System

We study a controllable two‐station tandem queueing system, where customers (jobs) must first be processed at upstream station and then the downstream station. A manager dynamically allocates the service resource to each station to adjust the service rate, leading to a tradeoff between the holding cost and resource cost. The goal of the manager is to find the optimal policy to minimize the long‐run average costs. The problem is constructed as a Markov decision process (MDP). In this paper, we consider the model in which the resource cost and service rate functions are more general than linear. We derive the monotonicity of the optimal allocation policy by the quasiconvexity properties of the value function. Furthermore, we obtain the relationship between the two stations’ optimal policy and conditions under which the optimal policy is unique and has the bang‐bang control property. Finally, we provide some numerical experiments to illustrate these results.

Retrial tandem queue with controllable strategy of repeated attempts

We consider a tandem retrial queueing system with single-server first station (Station 1) and multi-server second station (Station 2). The input flow at Station 1 is described by a Batch Markovian Arrival Process. If arriving customers meet a busy server of Station 1, they go to the orbit and repeat their attempts to get service in a random time interval. The system has two operation modes which are distinguished by a strategy of repeated attempts. The mode changes dynamically depending on the number of customers in the orbit according to the threshold control policy. We calculate the stationary distribution of the system content at the embedded epochs and at an arbitrary time and derive formulas for the main performance measures. Using these formulas, we are able to calculate the value of the cost criterion which evaluates the quality of service in the system as a function of the threshold level. Numerical results illustrate the behavior of performance measures and the cost criterion as functions of the threshold.

A Common Model for the Approximate Analysis of Tandem Queueing Systems With Blocking

This work proposes a common model that unifies several approximate methods for the analysis of tandem queueing systems with blocking. Two abstract functions are introduced to characterize the relationship between starvation and blockage in neighboring machines. Using the functions, the equilibrium equations are determined, and a general procedure is developed to solve them. The sufficient conditions for the existence and uniqueness of a solution are obtained. More generally, the proposed model leads to the concept of a bidirectional cascade system, in which each subsystem has both forward and backward inputs and forward and backward outputs. Finally, the theory of the common model is applied to specific systems to demonstrate its applicability.

Dynamic control of a tandem system with abandonments

The goal of this paper is to provide a model that is an extension of classic scheduling problems for a tandem queueing system by including customer impatience. In such scenarios, the server(s) must balance abandonments from each phase of service with the need to prioritize higher reward customers. This presents an interesting challenge since the trade-off between the cost of abandonments and revenue maximization is not at all clear. As a result of customer abandonments, traditional solution techniques are not available. In particular, uniformization is not possible since the transition rates are unbounded. We do our analysis in continuous time, using the continuous-time Markov decision process framework to discern simple relationships of the value functions depending on the starting state. We then use sample path arguments to analyze the structure of the optimal policy. We provide conditions under which an optimal policy is nonidling and never splits servers (except to avoid idling). We then consider the single-server model as a proxy for the decision problem when there are more customers at each station than there are servers. In this case, we provide conditions under which it is optimal to prioritize each station.

Optimal control of queueing systems with non-collaborating servers

We study the dynamic server allocation problem for tandem queueing systems with an equal number of stations and servers. The servers are flexible, yet non-collaborative, so that at most one server can work at a station at any time. The objective is to maximize the long-run average throughput. We show that if each server is the fastest at one station, then a dedicated server assignment policy is optimal for systems of arbitrary size and with general service requirement distributions. Otherwise, the optimal policy is more complex as servers must divide their time between stations. For Markovian systems with two stations and two servers, we characterize the optimal policy completely. For larger Markovian systems, we use our results for two-station systems to propose four heuristic server assignment policies and provide computational results that show that our heuristics are near-optimal. We also compare collaborative and non-collaborative settings to evaluate the benefits of dynamic server allocation, as opposed to collaboration, in systems with flexible servers. We conclude that the loss in the long-run average throughput due to lack of collaboration is mitigated by the similarity of the tasks in the system, and cross-training can still be beneficial in non-collaborative systems.

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

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 back-order 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.

Priority tandem queueing system with retrials and reservation of channels as a model of call center

We study a tandem queueing system consisting of two multi-server stations and finite intermediate buffer. Customers arrive at the first station of the tandem according to a Markovian Arrival Process. The first station does not have a buffer. Customers, who do not succeed to enter the service immediately upon arrival, retry for the service after a random amount of time. After service completion at the first station, a customer leaves the system permanently or moves for the service at the second station. A part of customers entering the second station have a priority over other customers. The priority is provided by means of reservation of a part of servers for the service of priority customers only. Usually, non-priority customers are not allowed to occupy the reserved servers. However, if the queue of non-priority customers in the intermediate buffer becomes larger than some preassigned threshold while there are free servers, a non-priority customer is picked-up for the service. Customers staying in the buffer are impatient. Priority and non-priority customers have different patience time and may leave the system or return to the first station if the patience time expires. The system is analyzed in steady state. A condition for existence of the stationary regime in the system is derived, the steady state distribution and various performance measures of the system are calculated, some illustrative numerical examples are discussed. A tandem queue under consideration is suitable, e.g., for modeling call centers with Interactive Voice Response Machines. Analysis presented in this paper was implemented in borders of the applied project funded by one of the banks.

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.

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.

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.

Optimal control of noncollaborative servers in two‐stage tandem queueing systems

AbstractWe consider two‐stage tandem queueing systems with dedicated servers in each station and a flexible server that is trained to serve both stations. We assume no arrivals, exponential service times, and linear holding costs for jobs present in the system. We study the optimal dynamic assignment of servers to jobs assuming a noncollaborative work discipline with idling and preemptions allowed. For larger holding costs in the first station, we show that (i) nonidling policies are optimal and (ii) if the flexible server is not faster than the dedicated servers, the optimal server allocation strategy has a threshold‐type structure. For all other cases, we provide numerical results that support the optimality of threshold‐type policies. Our numerical experiments also indicate that when the flexible server is faster than the dedicated server of the second station, the optimal policy may have counterintuitive properties, which is not the case when a collaborative service discipline is assumed. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 435–446, 2014

Modeling Multi-node Network under Heterogeneous M2M Services Arrival

With the development of IoT (Internet of Things), M2M (Machine to Machine, Man to Machine, Machine to Mobile network) services have found an increasingly wide utilization in many fields. In this paper, we investigate the network performance of heterogeneous M2M services. The multi-node bearing network is modeled as a tandem queuing system. Superposition of multiple MMBP-2 is made to model the arrival process in order to describe the characteristics of mixed M2M traffic arrival. To simplify resolving procedure, we innovatively propose a mathematical method to synthesize N MMBP-2 as one MMBP2 N . Then solving a multi-input tandem queuing system is converted to solving that with an ordinary single input. A decomposition technique and a parameter fitting algorithm are used to solve the tandem queuing system in this paper. We explore network performance of heterogeneous M2M services under two simulation scenarios. Some reliable measures to guarantee QoS (Quality of Service) of the network are suggested in this paper.

DELAY IN A TANDEM QUEUEING MODEL WITH MOBILE QUEUES: AN ANALYTICAL APPROXIMATION

In this paper, we analyze the end-to-end delay performance of a tandem queueing system with mobile queues. Due to state-space explosion, there is no hope for a numerical exact analysis for the joint-queue-length distribution. For this reason, we present an analytical approximation that is based on queue-length analysis. Through extensive numerical validation, we find that the queue-length approximation exhibits excellent performance for light traffic load.

A tollbooth tandem queue with heterogeneous servers

We study a tandem queueing system with K servers and no waiting space in between. A customer needs service from one server but can leave the system only if all down-stream servers are unoccupied. Such a system is often observed in toll collection during rush hours in transportation networks, and we call it a tollbooth tandem queue. We apply matrix-analytic methods to study this queueing system, and obtain explicit results for various performance measures. Using these results, we can efficiently compute the mean and variance of the queue lengths, waiting time, sojourn time, and departure delays. Numerical examples are presented to gain insights into the performance and design of the tollbooth tandem queue. In particular, it reveals that the intuitive result of arranging servers in decreasing order of service speed (i.e., arrange faster servers at downstream stations) is not always optimal for minimizing the mean queue length or mean waiting time.

Estimation of Mold Remaining Duration Considering Reworks

Mold production process is fraught with uncertainty, especially the uncertainty of process and number of reworks which has made accurate estimates and control mold production cycle to become a worldwide problem. To address this issue, tandem and hybrid manufacturing queuing system considering reworks is used to presents the mold manufacturing process, respectively. Under the assumption that the orders are processing under the first come first served principal and both the normal processing and rework time are exponential distributed, the event-based state transition process of the manufacturing queuing system has been analyzed, meanwhile, the concept of the rework strength is proposed and the event interval is approximately exponential distribution. On that basis, the mold remaining duration approximate mathematical distribution model is deducted by combining the sum of independent gamma random variables. A comparative analysis of the numerical results and simulation results under different rework strength and initial states verified the reliability of proposed approximate model of remaining duration.