Pooling in tandem queueing networks with non-collaborative servers

This paper considers pooling several adjacent stations in a tandem network of single-server stations with finite buffers. When stations are pooled, we assume that the tasks at those stations are pooled but the servers are not. More specifically, each server at the pooled station picks a job from the incoming buffer of the pooled station and conducts all tasks required for that job at the pooled station before that job is placed in the outgoing buffer. For such a system, we provide sufficient conditions on the buffer capacities and service times under which pooling increases the system throughput by means of sample-path comparisons. Our numerical results suggest that pooling in a tandem line generally improves the system throughput—substantially in many cases. Finally, our analytical and numerical results suggest that pooling servers in addition to tasks results in even larger throughput when service rates are additive and the two systems have the same total number of storage spaces.     

On the order of tandem queues

We consider the optimal order of servers in a tandem queueing system withm stages, an unlimited supply of customers in front of the first stage, and a service buffer of size 1 but no intermediate storage buffers between the first and second stages. Service times depend on the servers but not the customers, and the blocking mechanism at the first two stages is manufacturing blocking. Using a new characterization of reversed hazard rate order, we show that if the service times for two servers are comparable in the reversed hazard rate sense, then the departure process is stochastically earlier if the slower server is first and the faster server is second than if the reverse is true. This strengthens earlier results that considered individual departure times marginally. We show similar results for the last two stages and for other blocking mechanisms. We also show that although individual departure times for a system with servers in a given order are stochastically identical to those when the order of servers is reversed, this reversibility property does not hold for the entire departure process.     

Optimal server assignment in a two-stage tandem queueing system

We study Markovian queueing systems consisting of two stations in tandem. There is a dedicated server in each station and an additional server that can be assigned to any station. Assuming that linear holding costs are incurred by jobs in the system and two servers can collaborate to work on the same job, we determine structural properties of optimal server assignment policies under the discounted and the average cost criteria.     

Optimal assignment of servers to tasks when collaboration is inefficient

Consider a Markovian system of two stations in tandem with finite intermediate buffer and two servers. The servers are heterogeneous, flexible, and more efficient when they work on their own than when they collaborate. We determine how the servers should be assigned dynamically to the stations with the goal of maximizing the system throughput. We show that the optimal policy depends on whether or not one server is dominant (i.e., faster at both stations) and on the magnitude of the efficiency loss of collaborating servers. In particular, if one server is dominant then he must divide his time between the two stations, and we identify the threshold policy the dominant server should use; otherwise each server should focus on the station where he is the faster server. In all cases, servers only collaborate to avoid idleness when the first station is blocked or the second station is starved, and we determine when collaboration is preferable to idleness as a function of the efficiency loss of collaborating servers.     

Approximation of throughput in tandem queues with multiple servers and blocking

In this paper, we develop an approximation method for throughput in tandem queues with multiple independent reliable servers at each stage and finite buffers between service stations. We consider the blocking after service (BAS) blocking protocol of each service stage. The service time distribution of each server is exponential. The approximation is based on the decomposition of the system into a set of coupled subsystems which are modeled by two-stage tandem queue with two buffers and are analyzed by using the level dependent quasi-birth-and-death (LDQBD) process.     

Server allocation for zero buffer tandem queues

In this paper we consider the problem of allocating servers to maximize throughput for tandem queues with no buffers. We propose an allocation method that assigns servers to stations based on the mean service times and the current number of servers assigned to each station. A number of simulations are run on different configurations to refine and verify the algorithm. The algorithm is proposed for stations with exponentially distributed service times, but where the service rate at each station may be different. We also provide some initial thoughts on the impact on the proposed allocation method of including service time distributions with different coefficients of variation.     

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.     

Allocation of jobs and identical resources with two pooling centers

We examine the resource allocation problem of partitioning identical servers into two parallel pooling centers, and simultaneously assigning job types to pooling centers. Each job type has a distinct Poisson arrival rate and a distinct holding cost per unit time. Each pooling center becomes a queueing system with an exponential service time distribution. The goal is to minimize the total holding cost. The problem is shown to be polynomial if a job type can be divided between the pooling centers, and NP-hard if dividing job types is not possible. When there are two servers and jobs cannot be divided, we demonstrate that the two pooling center configuration is rarely optimal. A heuristic which checks the single pooling center has an upper bound on the relative error of 4/3. The heuristic is extended for the multiple server problem, where relative error is bounded above by the number of servers.      

Tandem queue with group arrivals and no intermediate buffer

We study a system with two single-server stations in series. There is an infinite buffer in front of the first station and no buffer between the two stations. The customers come in groups; the groups contain random numbers of customers and arrive according to a Poisson process. Assuming general service time distributions at the two stations, we derive the Laplace transform and the recursive formula for the moments of the total time spent in the tandem system (waiting time in the system) by an arbitrary customer. From the Laplace transform, we conclude that the optimal order of the servers for minimizing the waiting time in the system does not depend on the group size.     

Admission control strategies for tandem Markovian loss systems

Motivated by communication networks, we study an admission control problem for a Markovian loss system comprised of two finite capacity service stations in tandem. Customers arrive to station 1 according to a Poisson process, and a gatekeeper, who has complete knowledge of the number of customers at both stations, decides whether to accept or reject each arriving customer. If a customer is rejected, a rejection cost is incurred. If an admitted customer finds that station 2 is full at the time of his service completion at station 1, he leaves the system and a loss cost is incurred. The goal is to find easy-to-implement policies that minimize long-run average cost per unit time. We formulate two intuitive, extremal policies and provide analytical results on their performances. We also present necessary and/or sufficient conditions under which each of these policies is optimal. Next, we show that for some states of the system it is always optimal to admit new arrivals. We also fully characterize the optimal policy when the capacity of each station is two and discuss some characteristics of optimal policies in general. Finally, we design heuristic admission control policies using these insights. Numerical experiments indicate that these heuristic policies yield near-optimal long-run average cost performance.     

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. AMS subject classification: 90B22, 90B36     

A scheduling problem for several parallel servers

In this paper, we study a scheduling problem of jobs from two different queues on several parallel servers. Jobs have exponentially distributed processing times, and incur costs per unit of time, until they leave the system, and there are no arrivals to the system at any time. The objective is to find the optimal strategy, i.e., to allocate the servers to the queues, such that the expected holding costs are minimized. We give a sufficient condition for which it is always optimal to allocate the servers only to jobs of a certain queue. Finally, the case of two servers is completely solved.     

Work-Conserving Tandem Queues

Consider a tandem queue of two single-server stations with only one server for both stations, who may allocate a fraction α of the service capacity to station 1 and 1−α to station 2 when both are busy. A recent paper treats this model under classical Poisson, exponential assumptions.Using work conservation and FIFO, we show that on every sample path (no stochastic assumptions), the waiting time in system of every customer increases with α. For Poisson arrivals and an arbitrary joint distribution of service times of the same customer at each station, we find the average waiting time at each station for α = 0 and α = 1. We extend these results to k ≥ 3 stations, sample paths that allow for server breakdown and repair, and to a tandem arrangement of single-server tandem queues.This revised version was published online in June 2005 with corrected coverdate     

A Tandem Queue with Coupled Processors: Computational Issues

In Resing and Örmeci [16] it is shown that the two-stage tandem queue with coupled processors can be solved using the theory of boundary value problems. In this paper we consider the issues that arise when calculating performance measures like the mean queue length and the fraction of time a station is empty. It is assumed that jobs arrive at the first station according to a Poisson process and require service at both stations before leaving the system. The amount of work that a job requires at each of the stations is an independent, exponentially distributed random variable. When both stations are nonempty, the total service capacity is shared among the stations according to fixed proportions. When one of the stations becomes empty, the total service capacity is given to the nonempty station. We study the two-dimensional Markov process representing the numbers of jobs at the two stations. The problem of finding the generating function of the stationary distribution can be reduced to two different Riemann-Hilbert boundary value problems, where both problems yield a complete analytical solution. We discuss the similarities and differences between the two problems, and relate them to the computational aspects of obtaining performance measures.     

A Priority Tandem Queue with No Intermediate Buffer

Priority queues are important in modelling and analysis of manufacturing systems, and computer and communication networks. In this paper, a priority tandem queueing system with two stations in series is studied. There is no intermediate buffer between the two stations, and the lack of buffers may cause blocking at the first station. K types of customers arrive at the system according to Poisson processes. The expected delay in the system for each type of customer is obtained when all the customers have the same service time distribution at the second station. Two cases are studied in detail when service times are either all exponentially distributed or all deterministic.     

Inventory based allocation policies for flexible servers in?serial systems

Motivated by an industry example, we study a two-station serial system in which we allocate flexible servers in order to maximize throughput. We investigate two cases which are different in the way that servers work together when at the same station; namely collaboratively or non-collaboratively. For the collaborative case we prove the optimal policy to be such that the servers work together at a single station at any point in time. In addition to the policy being state-dependent, it also follows a switching-curve structure. In the non-collaborative case, on the other hand, it may be optimal to allocate servers to different stations. Some numerical examples and results regarding policy assignments, switching curves, and system throughput are presented.     

Admission control of a service station when the arrivals are internal

All studies in the admission control of a service station make decisions at arrival epochs. When arrivals are internal and are rejected from a queue, the rejected jobs have to be routed to other stations in the system. However the system will not know whether a job will be admitted to a queue or not until its arrival epoch to that queue. Thus, the system has to react dynamically and agilely to the decisions made at a specific queue and may try several queues before finding a queue that admits the job. This paper remedies these difficulties by changing the decision epochs of the admission control from arrival epochs to departure epochs with the actions of switching (keeping) the arrival stream on or off. Thus upstream stations will have information on the admission status of their downstream stations all the time. It is proved that the optimal policy for this revised admission control system is of control limit type for an M/G/1 queue. Comparisons of the optimal values and optimal policies for the admission controls made at arrival epochs and at departure epochs are included in the paper.     

Tandem queues with subexponential service times and finite buffers

We focus on tandem queues with subexponential service time distributions. We assume that number of customers in front of the first station is infinite and there is infinite room for finished customers after the last station but the size of the buffer between two consecutive stations is finite. Using (max, +) linear recursions, we investigate the tail asymptotics of transient response times and waiting times under both communication blocking and manufacturing blocking schemes. We also discuss under which conditions these results can be generalized to the tail asymptotics of stationary response times and waiting times.     

Optimal use of excess capacity in two interconnected queues

We consider a two-stage tandem queueing network where jobs from station 1 join station 2 with a certain probability. Each job incurs a linear holding cost, different for each station. Each station is attended by a dedicated server, and there is an additional server that is either constrained to serve in station 1 or can serve in both stations. Assuming no switching or other operating costs for the additional server, we seek an allocation strategy that minimizes expected holding costs. For a clearing system we show that the optimal policy is characterized by a switching curve for which we provide a lower bound on its slope. We also specify a subset of the state space where the optimal policy can be explicitly determined.     

Profit maximization in flexible serial queueing networks

We analyze the tradeoff between efficiency and service quality in tandem systems with flexible servers and finite buffers. We reward efficiency by assuming that a revenue is earned each time a job is completed, and penalize poor service quality by incorporating positive holding costs. We study the dynamic assignment of servers to tasks with the objective of maximizing the long-run average profit. For systems of arbitrary size, structured service rates, and linear or nonlinear holding costs, we determine the server assignment policy that maximizes the profit. For systems with two stations, two servers with arbitrary service rates, and linear holding costs, we show that the optimal server assignment policy is of threshold type and determine the value of this threshold as a function of the revenue and holding cost. The threshold can be interpreted as the best possible buffer size, and hence our results prove the equivalence of addressing service quality via a holding cost and via limiting the buffer size. Furthermore, we identify the optimal buffer size when each buffer space comes at a cost. We provide numerical results that suggest that the optimal policy also has a threshold structure for nonlinear holding costs. Finally, for larger systems with arbitrary service rates, we propose effective server assignment heuristics.