Partial Pooling in Tandem Lines with Cooperation and Blocking

For a tandem line of finite, single-server queues operating under the production blocking mechanism, we study the effects of pooling several adjacent stations and the associated servers into a single station with a single team of servers. We assume that the servers are cross-trained (so that they can work at several different stations) and that two or more servers can cooperate on the same job. For such a system, we provide sufficient conditions on the service times and sizes of the input and output buffers at the pooled station under which pooling will decrease the departure time of each job from the system (and hence increase the system throughput). We also show that pooling decreases the total number of jobs in the system at any given time and the sojourn time of each job in the system if the departure time of each job from the system is decreased by pooling and there is an arrival stream at the first station. Moreover, we provide sufficient conditions under which pooling will improve the holding cost of each job in the system incurred before any given time, and extend our results to closed tandem lines and to queueing networks with either a more general blocking mechanism or probabilistic routing. Finally, we present a numerical study aimed at quantifying the improvements in system performance obtained through pooling and at understanding which stations should be pooled to achieve the maximum benefit. Our results suggest that the improvements gained by pooling may be substantial and that the bottleneck station should be among the pooled stations in order to obtain the greatest benefit. AMS subject classification: 90B22     

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.     

Flexible servers in tandem lines with setup costs

We study the dynamic assignment of flexible servers to stations in the presence of setup costs that are incurred when servers move between stations. The goal is to maximize the long-run average profit. We provide a general problem formulation and some structural results, and then concentrate on tandem lines with two stations, two servers, and a finite buffer between the stations. We investigate how the optimal server assignment policy for such systems depends on the magnitude of the setup costs, as well as on the homogeneity of servers and tasks. More specifically, for systems with either homogeneous servers or homogeneous tasks, small buffer sizes, and constant setup cost, we prove the optimality of “multiple threshold” policies (where servers’ movement between stations depends on both the number of jobs in the system and the locations of the servers) and determine the values of the thresholds. For systems with heterogeneous servers and tasks, small buffers, and constant setup cost, we provide results that partially characterize the optimal server assignment policy. Finally, for systems with larger buffer sizes and various service rate and setup cost configurations, we present structural results for the optimal policy and provide numerical results that strongly support the optimality of multiple threshold policies.     

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.     

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.     

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.     

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.     

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.     

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.     

Throughput maximization for two station tandem systems: a proof of the Andradóttir–Ayhan conjecture

We study a tandem queueing network with two stations, M heterogeneous flexible servers, and a finite intermediate buffer. The objective is to dynamically assign the servers to the stations in order to maximize the throughput of the system. The form of the optimal policy for M≤3 was derived in two previous papers. In one of those papers, Andradóttir and Ayhan (Operations Research 53:516–531, 2005) provide a conjecture on the form of the optimal policy for M≥4. We prove their conjecture in this paper, showing that the optimal policy is defined by monotone thresholds and the ratios of the service rates among the servers. For M>1, we also prove that the optimal policy always uses the entire intermediate buffer.     

On the optimal design of tandem queueing systems with finite buffers

We consider tandem queueing systems that can be formulated as a continuous-time Markov chain, and investigate how to maximize the throughput when the queue capacities are limited. We consider various constrained optimization problems where the decision variables are of one or more of the following types: (1) expected service times, (2) queue capacities, and (3) the number of servers at the respective stations. After surveying our previous studies of this kind, we open up consideration of three new problems by presenting some numerical results that should give some insight into the general form of the optimal design.     

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.      

Workforce cross-training decisions in field service systems with two job types

When job types are heterogeneous in a multi-server service system, pooling servers to reduce system delay requires cross-training. Managers should balance a reduction in customer waiting time with high service costs and possibly reduced server efficiency due to cross-training. In a field service system with two job types and a fixed number of servers, the determination of the mix of dedicated and cross-trained servers is a critical managerial decision. We were motivated by a real field service situation to study a model where the objective is to minimize the sum of the average service costs and the customer delay costs per unit time. We use simulation to investigate the impact of various system parameters such as the number of servers, server utilization, and server efficiency on the optimal workforce mix.     

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.     

The multiple server effect: Optimal allocation of servers to stations with different service‐time distributions in tandem queueing networks

Traditionally, studies on tandem queueing networks concentrate on systems with infinite buffers, exponential service times, and/or single servers where solutions are more tractable. Less research can be found on more general, less tractable systems. We examine multipleserver systems with finite buffers and nonexponential service times, studying the effects of coefficient of variation (cv) of the servicetime distribution on the throughput of these systems, where cv differs among stations.Starting with the single station, we examine the effects of cv and the number of servers at the station on the distribution of interdeparture times. This insight helps explain the differences in throughput seen in the single (fast) server vs. multiple (slow) server problem. These results, in turn, shed light on the server allocation problem when cv differs among stations. We present some observations, as well as the intuition behind them.     

Heavy-traffic limits for an infinite-server fork–join queueing system with dependent and disruptive services

We study an infinite-server fork–join queueing system with dependent services, which experiences alternating renewal service disruptions. Jobs are forked into a fixed number of parallel tasks upon arrival and processed at the corresponding parallel service stations with multiple servers. Synchronization of a job occurs when its parallel tasks are completed, i.e., non-exchangeable. Service times of the parallel tasks of each job can be correlated, having a general continuous joint distribution function, and moreover, the service vectors of consecutive jobs form a stationary dependent sequence satisfying the strong mixing (\(\alpha \)-mixing) condition. The system experiences renewal alternating service disruptions with up and down periods. In each up period, the system operates normally, but in each down period, jobs continue to enter the system, while all the servers will stop working, and services received will be conserved and resume at the beginning of the next up period. We study the impact of both the dependence among service times and these down times upon the service dynamics, the unsynchronized queueing dynamics, and the synchronized process, assuming that the down times are asymptotically negligible. We prove FWLLN and FCLT for these processes, where the limit processes in the FCLT possess a stochastic decomposition property and the convergence requires the Skorohod \(M_1\) topology.     

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.     

Dynamic server allocation for unstable queueing networks with flexible servers

This paper is concerned with the dynamic assignment of servers to tasks in queueing networks where demand may exceed the capacity for service. The objective is to maximize the system throughput. We use fluid limit analysis to show that several quantities of interest, namely the maximum possible throughput, the maximum throughput for a given arrival rate, the minimum arrival rate that will yield a desired feasible throughput, and the optimal allocations of servers to classes for a given arrival rate and desired throughput, can be computed by solving linear programming problems. We develop generalized round-robin policies for assigning servers to classes for a given arrival rate and desired throughput, and show that our policies achieve the desired throughput as long as this throughput is feasible for the arrival rate. We conclude with numerical examples that illustrate the points discussed and provide insights into the system behavior when the arrival rate deviates from the one the system is designed for.     

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     

Analysis of a Fork/Join Synchronization Station with Inputs from Coxian Servers in a Closed Queuing Network

Fork/join stations are commonly used to model the synchronization constraints in queuing models of computer networks, fabrication/assembly systems and material control strategies for manufacturing systems. This paper presents an exact analysis of a fork/join station in a closed queuing network with inputs from servers with two-phase Coxian service distributions, which models a wide range of variability in the input processes. The underlying queue length and departure processes are analyzed to determine performance measures such as throughput, distributions of the queue length and inter-departure times from the fork/join station. The results show that, for certain parameter settings, variability in the arrival processes has a significant impact on system performance. The model is also used to study the sensitivity of performance measures such as throughput, mean queue lengths, and variability of inter-departure times for a wide range of input parameters and network populations.