MAD Dispersion Measure Makes Extremal Queue Analysis Simple

A notorious problem in queueing theory is to compute the worst possible performance of the GI/G/1 queue under mean-dispersion constraints for the interarrival- and service-time distributions. We address this extremal queue problem by measuring dispersion in terms of mean absolute deviation (MAD) instead of the more conventional variance, making available methods for distribution-free analysis. Combined with random walk theory, we obtain explicit expressions for the extremal interarrival- and service-time distributions and, hence, the best possible upper bounds for all moments of the waiting time. We also obtain tight lower bounds that, together with the upper bounds, provide robust performance intervals. We show that all bounds are computationally tractable and remain sharp also when the mean and MAD are not known precisely but are estimated based on available data instead. Summary of Contribution: Queueing theory is a classic OR topic with a central role for the GI/G/1 queue. Although this queueing system is conceptually simple, it is notoriously hard to determine the worst-case expected waiting time when only knowing the first two moments of the interarrival- and service-time distributions. In this setting, the exact form of the extremal distribution can only be determined numerically as the solution to a nonconvex nonlinear optimization problem. Our paper demonstrates that using mean absolute deviation (MAD) instead of variance alleviates the computational intractability of the extremal GI/G/1 queue problem, enabling us to state the worst-case distributions explicitly.

On the Benefits of Providing Timely Information in Ticket Queues with Balking and Calling Times

We study a phenomenon causing server time loss in ticket queues with balking and calling time. A customer who balks from the queue after printing a ticket leaves a virtual entity in the queue that requires server time to be cleared. The longer the queue, the larger the proportion of customers abandoning their place, and the larger the server time loss due to calling customers that left the queue. The solution is suggested by giving the customer the best possible estimate of her expected waiting time before printing a ticket, thus ensuring that, if she balks, no number in the queue is created that will waste server time. Although partially observable ticket queues have been studied in the literature, the addition of a calling time for absent customers creates a new type of problem that has been observed in real life but has not been formally addressed yet. We analyze this stochastic system, formulate its steady state probabilities, and calculate the system’s performance measures. The analytical solution provided here is robust and can be applied to a wide range of customers’ behavior functions. Finally, numerical analysis is performed that demonstrates the benefits of providing timely information to customers for different levels of traffic congestion.

An Optimization Model to Address Overcrowding in Emergency Departments Using Patient Transfer

Overcrowding of emergency departments (EDs) is a problem that affected many hospitals especially during the response to emergency situations such as pandemics or disasters. Transferring nonemergency patients is one approach that can be utilized to address ED overcrowding. We propose a novel mixed-integer nonlinear programming (MINLP) model that explicitly considers queueing effects to address overcrowding in a network of EDs, via a combination of two decisions: modifying service capacity to EDs and transferring patients between EDs. Computational testing is performed using a Design of Experiments to determine the sensitivity of the MINLP solutions to changes in the various input parameters. Additional computational testing examines the effect of ED size on the number of transfers occurring in the system, identifying an efficient frontier for the tradeoff between system cost (measured as a function of the service capacity and the number of patient transfers) and the systemwide average expected waiting time. Taken together, these results suggest that our optimization model can identify a range of efficient alternatives for healthcare systems designing a network of EDs across multiple hospitals.

An Integrated Bus Holding and Speed Adjusting Strategy Considering Passenger’s Waiting Time Perceptions

To solve the problems of bus bunching and large gaps, this study combines bus holding and speed adjusting to alleviate them respectively considering the characteristics of passenger’s perceived waiting time. The difference between passenger’s perceived waiting time at stops and actual time is described quantitatively through the expected waiting time of passengers. Bus holding based on a threshold method is implemented at any stops for bunching buses, and speed adjusting based on a Markovian decision model is implemented at limited stops for lagging buses. Simulations based on real data of a bus route show that the integrated control strategy is able to improve the service reliability and to decrease passengers’ perceived waiting time at stops. Several insights have been uncovered through performance analysis: (1) The increase of holding control strength results in improvement of the headway regularity, and leads to a greater perceived waiting time though; (2) Compared to traveling freely, suitable speed guidance will not slow down the average cruising speed in the trip; (3) The scale of passenger demand and through passengers are the two key factors influencing whether a stop should be selected as a speed-adjusting control point.

Particle Swarm Optimization and Maximum Entropy Results for MX/G/1 Retrial G-Queue with Delayed Repair

This paper examines an MX/G/1 retrial model with negative customers including the concepts of working vacation, Bernoulli feedback, delayed repair, state-dependent and multi-optional services. Such a queue is quite relevant in real world, for instance in computer systems, manufacturing organisations, packet-switching networks, telecommunication systems, etc. The arrival pattern of customers is according to Poisson distribution. The service is such that first essential service (FES) is provided to every customer and second optional service (SOS) is provided in k phases to those who wants to opt for the same. The negative customers may arrive during the time when the server is busy in serving a positive customer. This leads to breakdown of the server and thus the server has to be restored (repair) by the repair man. Some moments may be taken by the repair man to initiate the repair process, leading to delay in repair. In our work, firstly we have calculated performance measures like long run probabilities and orbit size along with some reliability indices. Then a relative study between the exact expected waiting time and approximate expected waiting time of the system is presented via maximum entropy approach. Also we perform cost optimization using particle swarm optimization (PSO method). Few numerical results are also provided.

Modelling producer behaviour in fixed route private transport services

Objectives: While fixed route private transport services have grown in prominence in the recent times, there is a dearth of models that specifically tackle pricing in it. The current study aims to model producer behaviour in this model, keeping in mind its peculiar physical characteristics. Methods: This study develops a rational-actor model of the behaviour of producers operating in this market. There is however, an added assumption of the heuristic of least perceptible difference to add behavioural realism to the model. Results: The predictions derived from the model developed in this study include repeated usage of a single type of round-trip for a non-zero interval of time, the convexity of expected-waiting time with respect to changes in prices and a negative relation between external (exogenous) demand at one-point of a path and the price charged at the other. Conclusion: Pricing in this market, due to physical factors, can exhibit unique features as modelled. Keywords: Private Transport; transport Pricing; producer behaviour in transport

Determination of Optimal Number of Servers at Network Queuing Nodes to Reduce Waiting Time in a Tertiary Institution Clinic in Bida, Nigeria

In this paper, a network queuing model that determines optimal numbers of servers at the nodes of the school clinic network queuing system to reduce waiting time of the patients has been presented. The relevant data was collected for a period four weeks, through direct observations and interviews. The number of arrivals and departures were also obtained. The total expected waiting time of the patient in the current system before modification was 50minutes with total number of 10 servers in all the nodes, while the total new expected waiting time of patient in the system after modification was reduced to 19 minutes with total number of 17 servers in all the nodes. The study has determined optimal number of servers at the nodes of the school clinic network system. Results from this study is an important information to the management of the school clinic for proper planning and better service delivery. Keywords: Network Queuing System, Nodes, Servers, School Clinic.

Concurrent sequences of Bernoulli trials

Probability and expectation are two distinct measures, both of which can be used to indicate the likelihood of certain events. However, expectation values, which are often associated with waiting times for success, may, at times, speak more clearly and poignantly about the uncertainty of an event than a theoretical probability. To illustrate the point, suppose the probability of choosing a winning lottery ticket is 2.5 × 10−8. This information may not communicate the unlikely odds of winning as clearly as a statement like, “If five lottery tickets are purchased per day, the expected waiting time for a first win is about 22000 years.”