Expected Queue Length Research Articles

Transient staffing at the beginning of work

Production and service systems often rely on stationary queueing formulas to determine the number of agents that need to be scheduled. However, this results in over-staffing decisions when the system starts empty. To address this issue, we propose a solution called transient staffing, which involves providing appointment times for agents to ensure that the expected queue length remains below a certain threshold. We model the system as a transient Erlang-A queue. To compute the appointment times, we employ a uniformized approximation that discretizes the elapsed time, which can be made as accurate as desired. We prove that the appointment times decrease with the arrival rate and traffic intensity, while increasing with the abandonment rate. Through numerical investigations, we find that adopting a transient staffing strategy leads to substantial cost savings, especially in scenarios where demand is high, customers are patient, traffic intensity is low, and the service level objective is intermediate. Furthermore, we use a fluid approximation to derive closed-form expressions for the agents' appointment times, and prove the impact of the system parameters. We show that this approximation performs well in predicting the first appointment times in contexts with high demand, low traffic intensity, and lax service level objective.

Dual core processors: Coupled queues: Transient performance evaluation

High-Performance Computing (HiPC) systems routinely employ multi/many – core processors. Specifically, dual – core processors find many applications in pervasive computing devices. Dual–core processors employ buffers for queueing the incoming jobs. Traditionally, the queues at the processors are assumed to be independent and the queueing system is analyzed in equilibrium for tractability purposes. Queues are modeled using Continuous Time Markov Chains (CTMC's) and the equilibrium performance measures are determined to analyze as well as design the queueing systems. In most interesting cases, the incoming jobs are routed to the queues using the Join the Shortest Queue (JSQ) policy. Thus, with such an adaptive routing algorithm, the two queues are evidently coupled and are not statistically independent. Hence traditional equilibrium performance evaluation doesn't provide realistic performance measures. In this research paper, the two queues associated with buffers in dual-core processors are considered to be coupled. The Coupled Queues are modeled using a Quasi – Birth – and – Death (QBD) process. Using traditional results related to QBD processes, equilibrium performance measures are determined. More interestingly, we demonstrate the tractability of the computation of transient probability distribution of a QBD process. In the research literature, transient analysis of the QBD process was shown to be tractable in the Laplace transform domain. But in this research paper, we prove that the matrix exponential eQt arising in transient analysis (where Q is the generator matrix of the QBD process) can be computed directly in the time domain rendering efficient transient analysis of QBD process. Using the transient probability mass function of queue length, estimation of transient performance measures such as expected queue length, average delay, and tail distribution can be determined. Further, optimal adaptive routing algorithms for coupled queues can be designed.

Congestion Problem during Covid-19 in the University College Hospital, Ibadan, Oyo State, Nigeria: An Application of Queuing Theory

Outpatient services have become an important component of healthcare especially during the era of pandemic (Covid-19). The amount of time a patient spends waiting in an outpatient setting may have an impact on their satisfaction with the service(s) provided. In this research, an attempt was made to investigate the arrival and service rates of patients for the period of one month between the hours of 9:00am and 2:00pm at the University College Hospital (UCH), Ibadan during Covid-19 using Poisson Process in order to deduce traffic intensity (ρ), expected number of patients at a steady rate (L), expected queue length (Lq), expected waiting time (Wq), total patients waiting (W) and mean number of Patients in the system. Results showed that more patients visit the clinic everyday of the week due to the high traffic intensity close to 100% and the high minutes spent in the hospital before being attended to. Conclusively, waiting time on Saturdays and Tuesdays are much compared to the remaining days.

Railcar reallocation optimization on water-rail network under uncertain busyness

Water-rail intermodal transportation can reduce cargo losses and transportation transferring costs. However, the imbalance between the capacity of the scheduled railway network and the large container freight demand greatly reduces operational efficiency. To minimize the total transportation cost and relocation cost, a railcar reallocation stochastic optimization model is formulated to deal with uncertain congestion in the railway network. To capture the uncertain busyness and queuing pattern, a hypercube spatial queue model is embedded in the optimization model by estimating the expected queue length and waiting time. To solve the proposed nonlinear nonconcave stochastic model, an approximate hypercube based iterative algorithm is proposed. A real-world case study is presented to show the effectiveness and efficiency of the proposed method. The proposed model outperforms the comparable deterministic model in the objective value. Sensitivity analyses on the ratio of the unit waiting cost and the unit travel cost for empty cars, and the total number of freight cars show the robustness of the proposed method.

Low-complexity scheduling algorithms with constant queue length and throughput guarantees

Distributed scheduling algorithms based on carrier sense multiple access (CSMA) are optimal in terms of the throughput and the steady-state queue lengths. However, they take a prohibitively long time to reach the steady-state, often exponential in the network size. Therefore for large networks that operate over a finite time horizon, apart from the guarantees on the steady-state queue lengths, performance guarantees on the short-term (i.e., transient) queuing behaviour are also required. To that end, we propose distributed scheduling algorithms that are guaranteed to have O(1) expected queue lengths not just in the steady-state but at every time instant, where O(⋅) is with respect to the network size. Further, our algorithms have O(1) complexity and support a constant fraction of the maximum throughput for typical wireless topologies. The central idea of our algorithms is to resolve collisions among pairs of conflicting nodes by assigning a master–follower hierarchy. The master–follower hierarchy can either be chosen randomly or based on the topology of the conflict graph, leading to different performance guarantees.In addition to these hierarchical collision resolution algorithms, which are primarily designed for the conflict graph-based interference model, we also propose an Aloha-based algorithm for the K-neighbour collision tolerance interference model, which is a generalization of the conflict graph model. We show that the proposed Aloha-based algorithm supports a constant fraction of the maximum throughput for typical wireless topologies.

Balancing admission control, speedup, and waiting in service systems

Admission control and service rate speedup may be used during periods of congestion to minimize customer waiting in different service settings. In a healthcare setting, this can mean sending patients to alternative care facilities that may take more time and/or provide less ideal treatment. While waiting can be detrimental to patient outcomes, strategies used to control congestion can also be costly. In this work, we examine a multi-server queueing system that considers both admission control and speedup. We use dynamic programming to characterize properties of the optimal control and find that in some instances the optimal policy has a simple form of a threshold policy. Leveraging this insight, we examine a queueing system where speedup is used when the number of customers (patients) in the system exceeds some threshold and admission control is used when that number exceeds some (potentially different) threshold. Using a fluid model and a stochastic loss model, we develop a methodology to derive approximations for the probability that speedup will be applied, the probability that admission control will be applied and the expected queue length customers experience. We use the approximations as the basis for a greedy heuristic to derive a near optimal solution to the original stochastic optimization problem. We use simulation to demonstrate the quality of these approximations and find that they can be quite accurate and robust. This analysis can provide insight to managers deciding how to balance admission control and speedup in service settings: when and to what extent to use each.

A single server queuing system with correlated reneging and feedback of served customers

In this article, we consider a single server queuing model with correlated reneging and feedback of served customers. The steady-state solution of the model is obtained using the matrix-decomposition method. The transient analysis of the model is also performed with the help of the Runge-Kutta method. Important measures of performance like expected queue length and expected waiting time in the queue are computed. Some comparisons are performed, and finally, the sensitivity analysis of the model is also carried out.

A simulation optimization method for deep-sea vessel berth planning and feeder arrival scheduling at a container port

Vessels served by a container port can usually be classified into two types: deep-sea vessels and feeders. While the arrival times and service times of deep-sea vessels are known to the port operator when berth plans are being devised, the service times of feeders are usually uncertain due to lack of data interchange between the port operator and the feeder operators. The uncertainty of feeder service times can incur long waiting lines and severe port congestion if the service plans for deep-sea vessels and feeders are poorly devised. This paper studies the problem of how to allocate berths to deep-sea vessels and schedule arrivals of feeders for congestion mitigation at a container port where the number of feeders to be served is significantly larger than the number of deep-sea vessels, and where the service times of feeders are uncertain. We develop a stochastic optimization model that determines the berth plans of deep-sea vessels and arrival schedules of feeders, so as to minimize the departure delays of deep-sea vessels and schedule displacements of feeders. The model controls port congestion through restricting the expected queue length of feeders. We develop a three-phase simulation optimization method to solve this problem. Our method comprises a global phase, a local phase, and a clean-up phase, where the simulation budget is wisely allocated to the solutions explored in different phases so that a locally optimal solution can be identified with a reasonable amount of computation effort. We evaluate the performance of the simulation optimization method using test instances generated based on the operational data of a container port in Shanghai.

Bayesian estimation of the expected queue length of a system M/M/1 with certain and uncertain priors

One of characteristics of the traffic intensity in M/M/1 queue is the expected queue length. In this article, the problem of Bayes, E-Bayes, and robust Bayes estimation of the expected length in a M/M/1 queue system under a scale invariant squared error loss function is considered. We find the Bayes, E-Bayes, and robust Bayes estimators of the expected length. Finally, a simulation study is conducted for comparison of the proposed Bayes estimators.

Note: The Value and Cost of the Customer’s Waiting Time

The (relative) cost of the customer’s waiting time has long been used as a key parameter in queuing models, but it can be difficult to estimate. A recent study introduced a new queue characteristic, the value of the customer’s waiting time, which measures how an increase in the total customer waiting time reduces the servers’ idle time. This paper connects and contrasts these two fundamental concepts in the queuing literature. In particular, we show that the value can be equal to the cost of waiting when the queue is operated at optimal. In this case, we can use the observed queue length to compute the value of waiting, which helps infer the cost of waiting. Nevertheless, these two measures have very different economic interpretations, and in general, they are not equal. For nonoptimal queues, comparing the value with the cost helps shed light on the underlying causes of the customer’s waiting. Although it is tempting to conclude that customers in a queue with a lower value of waiting expect to wait longer, we find that the value of waiting in general does not have a monotonic relationship with the expected waiting time, nor with the expected queue length.

Learning Unknown Service Rates in Queues: A Multiarmed Bandit Approach

Traditional scheduling problems in stochastic queueing systems assume that the statistical parameters are known a priori. In ''Learning unknown service rates in queues: A multiarmed bandit approach'', Krishnasamy, Sen, Johari, and Shakkottai consider the problem of online scheduling in a parallel-server system when the statistical parameters are unknown. They study this question in the stochastic multiarmed bandits framework with the queue length as the performance objective. In contrast to the classic stochastic multiarmed bandits problem, where the regret scales logarithmically with time, the authors show that the queue regret (difference in expected queue length between a bandit algorithm and a genie policy) exhibits a more complex behavior. It grows logarithmically in the initial stage and eventually decays almost inversely with time. This remarkable behavior is explained through the analysis of regenerative cycle lengths, which shorten with time as the bandit algorithm learns to stabilize the queues.

Batch service systems with heterogeneous servers

Bulk-service multi-server queues with heterogeneous server capacity and thresholds are commonly seen in several situations such as passenger transport or package delivery services. In this paper, we develop a novel decomposition-based solution approach for such queues using arguments from renewal theory. We then obtain the distribution of the waiting time measure for multi-type server systems. We also obtain other useful performance measures such as utilization, expected throughput time, and expected queue lengths.

On Scheduling Policies With Heavy-Tailed Dynamics in Wireless Queueing Systems

This paper takes a system view and studies a wireless queueing system where heavy-tailness may occur both at the traffic arrival and in the form of the multi-user interference. With the rapid development of AI technologies, this heavy-tailed traffic model has become more prevalent in the current network system, such as the file or data size used in the deep learning algorithm. We first re-visit the standard asymmetric queueing system with a mix of heavy-tailed and light-tailed traffic, but under a new variable-rate service model that not only better models the dynamics of the wireless medium but also includes the previous models as special cases. We then focus on the scheduling problem when heavy-tailed interference disrupts the serving link. The performance of queueing policies is investigated during an ON/OFF renewal channel process with heavy-tailed OFF periods, and the expected queue length and the throughput characteristic is studied under the priority as well as max-weight scheduling policies. The results show that the expected queue length of the heavy queue cannot be maintained as finite even under the most favorable priority policy. On the other hand, a priority policy can guarantee the finiteness of an expected queue length for the light queue, but the system is not throughput optimal any longer. It is further shown that no benefit can be provided by the max-weight scheduling policy to the light queue for the queue length behavior in a steady-state, though the system is always throughput optimal.

Pooling and Balking: Decisions on COVID-19 Testing

The most important issue in the prevention and mitigation of the COVID-19 pandemic is, perhaps, testing. To increase the convenience and efficiency of testing, many countries have set up walk-in testing sites. One fundamental decision for local governments or authorities to make is, should they build fewer centralized walk-in or drive-thru testing site (with more manpower and therefore a higher service rate) or more decentralized testing sites (each with less manpower and therefore a lower service rate)? To make this decision, one needs to understand the benefits and/or drawbacks of pooling incorporating people's balking behavior. In this short paper, we build classic queueing models to analytically characterize how system performance measures such as the expected waiting time, the expected queue length, and the balking probability change with the service rate, under different values of the implied utilization. These performance measures not only indicate the productivity and efficiency of testing operations, but also influence the safety and well-being of people and society.

Quasi-birth and death processes of two-server queues with stalling

This paper investigates an optimal K-policy for a two-server Markovian queueing system $$M/(M_1,M_2)/2/(B_1,B_2),$$ with one fast server $$S_1$$ and one slow server $$S_2$$ , using the matrix analytic method. Two buffers $$B_1$$ and $$B_2$$ are organized to form waiting lines of customers in which, buffer $$B_1$$ is of finite size $$K(< \infty )$$ and buffer $$B_2$$ is of infinite capacity. Buffer $$B_1$$ stalls customers who arrive when the system size (queue + service) is less than $$(K+1)$$ and dispatches a customer to the fast server $$S_1$$ only after $$S_1$$ completes its previous service. This K-policy is of threshold type which deals with controlling of informed customers and hence the customers have better choice of choosing the fast server routing through the buffer $$B_1$$ . The $$(K+2)$$ -nd customer who arrives when the number of customers present in the system is exactly $$(K+1)$$ has the Hobson’s choice of getting service from the slow server $$S_2.$$ Buffer $$B_2$$ accommodates other customers who arrive when the number of customers present in the system is $$(K+2)$$ or more and feeds them one after another to either buffer $$B_1$$ or the sever $$S_2$$ whichever event can first accept the customer at the head-of-the-line in $$B_2$$ . Queue length processes of interest are (1) $$q_1=\lim \limits _{t\rightarrow \infty }X_1(t)$$ and (2) $$q_2=\lim \limits _{t\rightarrow \infty }X_2(t)$$ , where $$X_1$$ (t) represents the number of customers who are in the buffers $$B_1$$ and $$B_2$$ and also in the service with server $$S_1$$ at time ‘t’ and $$X_2$$ (t) represents the number of customers available with server $$S_2$$ only. The bi-variate random sequence $$\mathbf{X}(t)=(X_1(t),X_2(t))$$ of the system size (queue $$+$$ service) forms a quasi-birth and death process (QBD). Steady state characteristics, and some of the performance measures such as the expected queue length, the probability that each server is busy etc are obtained. Numerical illustrations are provided based on the average cost function to explore the methodology of finding the best K-policy which minimizes the mean sojourn time of customers.

An exact root-free method for the expected queue length for a class of discrete-time queueing systems

For a class of discrete-time queueing systems, we present a new exact method of computing both the expectation and the distribution of the queue length. This class of systems includes the bulk-service queue and the fixed-cycle traffic-light (FCTL) queue, which is a basic model in traffic-control research and can be seen as a non-exhaustive time-limited polling system. Our method avoids finding the roots of the characteristic equation, which enhances both the reliability and the speed of the computations compared to the classical root-finding approach. We represent the queue-length expectation in an exact closed-form expression using a contour integral. We also introduce several realistic modifications of the FCTL model. For the FCTL model for a turning flow, we prove a decomposition result. This allows us to derive a bound on the difference between the bulk-service and FCTL expected queue lengths, which turns out to be small in most of the realistic cases.

Analysis of batch arrival bulk service queue with additional optional service multiple vacation and setup time

This paper studies an M[x]/G(a, b)/1 queuing system with additional optional service, multiple vacation and setup time. After completing the first service, the customers may opt for the second service with probability α or leave the system with probability 1 ‒ α. After completing a bulk service, if the queue size is less than 'α', then the server leaves for a vacation of random length. When he returns from the vacation, if the queue length is still less than 'α', he leaves for another vacation and so on. This process continues until he finds at least 'α' customer in the queue. After a vacation, if the server finds at least 'α' customer waiting for service, he requires a setup time 'G' to start the service. After this setup, he serves a batch of ξ customers (a ≤ ξ ≤ b). Using supplementary variable technique, the probability generating function (PGF) of the queue size, expected queue length, expected waiting time, expected busy period and expected idle period are derived. Numerical illustrations are presented to visualise the effect of system parameters.

The Value of the Customer's Waiting Time for General Queues*

ABSTRACTWe have developed a measure of the value of the customer's waiting time that is applicable to all queuing systems. Since the birth of the modern queuing theory over 100 years ago, this measure is the first addition to the list of the measures of performance of general queues that includes the servers’ utilization factor, the expected queue length, the expected waiting time, and some variations of the last two. The curves for trade‐offs between the servers’ utilization factor and the value of the customer's time can be used to supplement or replace the curves for trade‐offs between the servers’ utilization factor and the customer's expected queue length (or expected waiting time) that have been a fundamental part of the modern queuing theory. The decisions made with the value of the customer's waiting time will mirror the decision maker's goals more closely than the decisions made with the customer's expected queue length or expected waiting time that are surrogates for the value of the customer's waiting time. Although our definition of the value of the customer's time is deceptively simple, its implications can be significant and far reaching. It could change the way we pursue research in the queuing theory, the way we teach the queuing theory, and the way we design queuing systems in practice.

Customer and Cost Sharing in a Jackson Network

In this paper we consider a Jackson network of independent service stations. To improve upon the total expected queue length, the stations may redistribute the total arrival rate of customers over all queues. We investigate this situation by means of arrival rate games. These are cooperative TU cost games in which the stations are the players and the total cost of a coalition is linear in the total expected queue length. We show that these games are totally balanced; cooperation is beneficial to all stations. Further, we provide a tailor-made cost allocation rule that belongs to the core of the game.

A queuing model for CPU functional unit and issue queue configuration

In a superscalar processor, instructions of various types flow through an execution pipeline, traversing hardware resources which are mostly shared among many different instruction types. A notable exception to shared pipeline resources is the collection of functional units, the hardware that performs specific computations. In a trade-off of cost versus performance, a pipeline designer must decide how many of each type of functional unit to place in a processor’s pipeline. In this paper, we model a superscalar processor’s issue queue and functional units as a novel queuing network. We treat the issue queue as a finite-sized waiting area and the functional units as servers. In addition to common queuing problems, customers of the network share the queue but wait for specific servers to become ready (e.g., addition instructions wait for adders). Furthermore, the customers in this queue are not necessary ready for service, since instructions may be waiting for operands. In this paper we model a novel queuing network that provides a solution to the expected queue length of each type of instruction. This network and its solution can also be generalized to other problems, notably other resource-allocation issues that arise in superscalar pipelines.