Search

The workload of a generalized n-site asymmetric simple inclusion process (ASIP) is investigated. Three models are analyzed. The first model is a serial network for which the steady-state Laplace–Stieltjes transform (LST) of the total workload in the first k sites ( k ≤ n ) just after gate openings and at arbitrary epochs is derived. In a special case, the former (just after gate openings) turns out to be an LST of the sum of k independent random variables. The second model is a 2-site ASIP with leakage from the first queue. Gate openings occur at exponentially distributed intervals, and the external input processes to the stations are two independent subordinator Lévy processes. The steady-state joint workload distribution right after gate openings, right before gate openings and at arbitrary epochs is derived. The third model is a shot-noise counterpart of the second model where the workload at the first queue behaves like a shot-noise process. The steady-state total amount of work just before a gate opening turns out to be a sum of two independent random variables. [ABSTRACT FROM AUTHOR]

This paper considers an unobservable two-site tandem queueing system attended by an alternating server. We study the strategic customer behaviour under two threshold-based operating policies, applied by a profit-maximizing server, while customers' strategic behaviour and server's switching costs are taken into account. Under the Exact-N policy, in each cycle the server first completes service of N customers in the first stage ( Q 1 ), then switches to the second stage ( Q 2 ) and then serves those N customers before switching back to Q 1 to start a new cycle. This policy leads to a mixture of Follow-the-Crowd and Avoid-the-Crowd customer behaviour. In contrast, under the N-Limited policy, the server switches from Q 1 to Q 2 also when the first queue is emptied, making this regime work-conserving and leading only to Avoid-the-Crowd behaviour. Performance measures are obtained using matrix geometric methods for both policies and any threshold N, while for sequential service ( N = 1 ) explicit expressions are derived. It is shown that the system's stability condition is independent of N, and of the switching policy. Optimal performances in equilibrium, under each of these switching policies, are analysed and compared through a numerical study. [ABSTRACT FROM AUTHOR]

We study a Markovian single-server ticket queue where, upon arrival, each customer can draw a number from a take-a-number machine, while the number of the customer currently being served is displayed on a panel. The difference between the above two numbers is called the "virtual queue length." We consider a nonhomogeneous population of customers comprised of two types: "regular" and "strategic." Upon arrival, a regular customer, regardless of the value of the virtual queue length, draws a number from the machine, joins the queue and waits in the system until being served. A strategic customer, depending on the virtual queue length, may either join, leave, or go to "orbit" for a random duration. If, upon return from orbit, a strategic customer realizes that s/he missed her/his turn, s/he balks. Otherwise, s/he joins the queue and waits to be served. We analyze this intricate stochastic system, calculate its steady-state probabilities, derive the sojourn time's Laplace–Stieltjes transform of a regular and of a strategic customer and calculate the system's performance measures. Finally, an economic analysis is performed to determine the optimal mean orbiting time of strategic customers for two types of objective functions. Numerical examples are presented. [ABSTRACT FROM AUTHOR]

We propose a novel approach, based on a Human Leukocyte Antigen (HLA) best-fit criterion, to dynamically allocate live organs (specifically, kidneys) to candidates needing transplantation. A ‘reward’ is assigned to each level of HLA fit, such that higher rewards are attributed to transplants between better-matched candidates and kidneys. We also envision future technologies by which it will be possible to store organs so that two queues may form: waiting candidates or stored kidneys. Consequently, a double-ended queue of candidates and kidneys is constructed, where the lifetime of a stored kidney is random, and candidates queueing for transplantation may die (‘renege’) while waiting. We derive expressions for the probability that a candidate gets a kidney before reneging; for the mean numbers of waiting candidates and of stored kidneys; and for a candidate's or kidney's mean sojourn time. Assuming a best-HLA-fit matching policy, we study three measures of effectiveness: (i) Rate of Reward from Transplantation (RRT); (ii) Expected Reward per Transplantation (ERT), calculated as RRT divided by the rate of performed transplantations, and (iii) Gained rate of reward per one dollar of expenditure. The optimal fraction of kidneys that should be stored so as to maximize the rate of reward per one dollar of expenditure is numerically determined. [ABSTRACT FROM AUTHOR]

We consider an asymmetric inclusion process, which can also be viewed as a model of n queues in series. Each queue has a gate behind it, which can be seen as a server. When a gate opens, all customers in the corresponding queue instantaneously move to the next queue and form a cluster with the customers there. When the nth gate opens, all customers in the nth site leave the system. For the case where the gate openings are determined by a Markov renewal process, and for a quite general arrival process of customers at the various queues during intervals between successive gate openings, we obtain the following results: (i) steady-state distribution of the total number of customers in the first k queues, $$k=1,\dots ,n$$ ; (ii) steady-state joint queue length distributions for the two-queue case. In addition to the case that the numbers of arrivals in successive gate opening intervals are independent, we also obtain explicit results for a two-queue model with renewal arrivals. [ABSTRACT FROM AUTHOR]

Two independent Poisson streams of jobs flow into a single-server service system having a limited common buffer that can hold at most one job. If a type- $$i$$ job ( $$i=1,2$$ ) finds the server busy, it is blocked and routed to a separate type- $$i$$ retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. We study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a Riemann-Hilbert boundary value problem. Various performance measures are calculated and numerical results are presented. In particular, numerical results demonstrate that the proposed multiple access system with two types of jobs and constant retrial rates provides incentives for the users to respect their contracts. [ABSTRACT FROM AUTHOR]

Exact analysis of a multi-server Markovian queueing system with cross selling in steady-state is presented. Cross selling attempt is initiated at the end of a customer's service every time the number of customers in the system is below a threshold. Both probability generating functions (PGFs) and matrix geometric methods are employed. The relation between the methods is revealed by explicitly calculating the entries of the matrix geometric rate-matrix R. Those entries are expressed in terms of the roots of a determinant of a matrix related to the set of linear equations involving the PGFs. This is a further step towards understanding of the analytical relationship between the two methods. Numerical results are presented, showing the effect of the cross selling intensity and of the threshold level on the systems performance measures. Finally, for a given set of parameters, the optimal threshold level is determined. [ABSTRACT FROM AUTHOR]

We study a single-server queue in which the service consists of two independent stages. The first stage is generic and can be performed even in the absence of customers, whereas the second requires the customer to be present. When the system is empty of customers, the server produces an inventory of first-stage (‘preliminary’) services (denoted PSs), which is used to reduce customers’ overall sojourn times. We formulate and analyze the queueing-inventory system and derive its steady-state probabilities by using the matrix geometric method, which is based on calculating the so called rate matrix R . It is shown that the system's stability is not affected by the production rate of PSs, and that there are cases in which utilizing the server's idle time to produce PSs actually increases the fraction of time during which the server is dormant. A significant contribution is the derivation of an explicit expression of R , whose entries are written in terms of Catalan numbers. This type of result is rare in the literature and enables large-scale problems to be solved with low computational effort. Furthermore, by utilizing Laplace–Stieltjes transform and its inverse, we obtain the distribution function of customers’ sojourn time. Finally, based on the probabilistic study, we carry out an economic analysis using a practical example from the fast food industry. [ABSTRACT FROM AUTHOR]

The risk of infection from the COVID-19 virus dictates businesses, such as supermarkets and department stores, to impose limits on the maximal number of customers allowed inside a store at any given time. These social distancing constraints generate long queues of waiting customers outside such businesses. This work investigates the impact of infection risk on arriving customers' strategic decisions regarding joining such queues. We consider a typical store where the floor is divided into two separate areas: (i) a shopping area with at most K shoppers allowed, and (ii) a payment area with c ≥ 1 parallel servers and an adjacent limited waiting space of size N ≥ 0. When the shopping area is full, a newly arriving customer observes only the outside queue and decides whether to join or balk. We investigate customers' individual joining strategies, as well as social optimization, with a utility function that takes into account not only the cost associated with waiting times (as in Naor's (1969) celebrated model), but also the cost related to the risk of infection. We propose an innovative risk measure that is a function of both the number of customers already in line, and those that a tagged customer 'meets' while waiting to enter the store. Consequently, expressions for mean waiting times and infection risk are derived and explicit formulas are obtained for limit values of the parameters. Our results can be used by authorities and customers alike to determine the maximal allowed queue sizes that ensure safety and reduce the risk of infection while minimizing associated costs., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2021 Elsevier Ltd. All rights reserved.)

The COVID-19 pandemic has forced numerous businesses such as department stores and supermarkets to limit the number of shoppers inside the store at any given time to minimize infection rates. We construct and analyze two models designed to optimize queue sizes and customer waiting times to ensure safety. In both models, customers arrive randomly at the store and, after receiving permission to enter, pass through two service phases: shopping and payment. Each customer spends a random period of time shopping (first phase) and then proceeds to the payment area of the store (second phase) where cashiers are assigned to serve customers. We propose a novel approach by which to calculate the risk of a customer being infected while queueing outside the store, while shopping, and while checking out with a cashier. The risk is proportional to the second factorial moment of the number of customers occupying the space in each phase of the shopping route. We derive equilibrium strategies for a Stackelberg game in which the authority acts as a leader who first chooses the maximum number of customers allowed inside the store to minimize the risk of infection. In the first model, store' management chooses the number of cashiers to provide to minimize its operational costs and its customers' implied waiting costs based on the number allowed in the store. In the second model, the store partitions its total space into two separate areas - one for shoppers and one for the cashiers and payers - to increase cashiers' safety. Our findings and analysis are useful and applicable for authorities and businesses alike in their efforts to protect both customers and employees while reducing associated costs., (© 2020 Elsevier Ltd. All rights reserved.)