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Throughout its history, Operational Research has evolved to include methods, models and algorithms that have been applied to a wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first summarises the up-to-date knowledge and provides an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion and used as a point of reference by a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes. [ABSTRACT FROM AUTHOR]

We consider a two-stage service system with two types of servers, namely subordinates who perform the first-stage service and supervisors who have their own responsibilities in addition to collaborating with the subordinates on the second-stage service. Rewards are earned when first- or second-stage service is completed and when supervisors finish one of their own responsibilities. Costs are incurred when impatient customers abandon without completing the second-stage service. Our problem is to determine how the supervisors should distribute their time between their joint work with the subordinates and their own responsibilities. Under the assumptions that service times at both stages are exponentially distributed and that the customers waiting for second-stage service abandon after an exponential amount of time, we prove that one of two policies will maximize the long-run average profit. Namely, it is optimal for supervisors to start collaborating with subordinates either when subordinates can no longer serve new customers or as soon as there is a customer ready for second-stage service. Furthermore, we show that the optimality condition is a simple threshold on the system parameters. We conclude by proving that pooling supervisors (and their associated subordinates) improves system performance, but with limited returns as more supervisors are pooled. [ABSTRACT FROM AUTHOR]

We explore settings where it is necessary (due to physical or operational constraints) or desirable (due to synergies or ease of implementation) to assign resources to tasks in a synchronous manner. We model the system as a queueing network with flexible servers and introduce the notion of a configuration to address the synchronous assignment of servers. This allows for a unified approach to determine the effects of resource synchronization, covering a wide range of problems in the literature. The maximal capacity of the system is given by the solution of a linear programming problem that also provides the optimal fractions of time the servers should spend in different configurations. This is used as a basis for constructing policies that have capacity arbitrarily close to the maximal capacity. We contrast synchronous server assignment with an asynchronous approach (focusing on independently scheduling individual servers rather than configurations) and show that synchronous server assignment is attractive with respect to applicability (it can capture constraints on server assignment and synergies among servers), implementation (it may have significantly fewer combinations of server allocations), and capacity (when both are applicable, asynchronous and synchronous server assignment will yield the same maximal capacity). Finally, we illustrate our modeling framework using several examples. [ABSTRACT FROM AUTHOR]

Standard server assignment policies for multi-server queueing stations include the noncollaborative policy, where the servers work in parallel on different jobs; and the fully collaborative policy, where the servers work together on the same job. However, if each job can be decomposed into subtasks with no precedence relationships, then we consider a form of server coordination named task assignment where the servers work in parallel on different subtasks of the same job. We identify the task assignment policy that maximizes the long-run average throughput of a queueing station with finite internal buffers when blocked servers can be idled or reassigned to either replace or collaborate with other servers on unblocked subtasks. We then compare the server coordination policies and show that the task assignment is best when the servers are highly specialized; otherwise, the fully collaborative or noncollaborative policies are preferable depending on whether the synergy level among the servers is high or not. We also provide numerical results that quantify our previous comparison. Finally, we address buffer allocation in longer lines where there are precedence relationships between some of the tasks, and present numerical results that suggest our comparisons for one queueing station generalize to longer lines. [ABSTRACT FROM AUTHOR]

Cross‐training is an important tool to improve the performance of manufacturing and service systems through dynamic worker allocation. This study investigates how cross‐training can be leveraged when worker collaboration is not possible and frequent worker reassignment is undesirable. We consider a tandem queueing system with finite buffers between tasks. We show that if each worker is the fastest at a different task, then the optimal policy is always dedicated, regardless of the magnitude of the reassignment costs. Otherwise, if the system is Markovian with two homogeneous tasks and a faster and a slower worker, we completely characterize how the profit‐optimal policy depends on the (constant) reassignment cost. We also prove that for any given reassignment cost, dedicated worker allocation policies are strictly suboptimal for large enough buffer sizes. Instead, the faster worker should move to the downstream task only if there are enough jobs in the intermediate buffer and return to the upstream task when the buffer is sufficiently empty. Furthermore, the benefit of task switching increases as the buffer size becomes larger. We use our theoretical results to develop worker allocation heuristics both for more general systems with two tasks and for systems with more tasks. Numerical experimentation provides insights on how the profit depends on the buffer sizes and reassignment costs, shows that policies that are optimal when workers are collaborative result in excessive switching for non‐collaborative workers, and indicates that our pick‐the‐best heuristics perform well in all settings. [ABSTRACT FROM AUTHOR]

10.1007/s11134-008-9078-5 13 Dai JG, He S. Many-server queues with customer abandonment: Numerical analysis of their diffusion model. Queueing systems with customer abandonments drew a lot of attention in the literature (see [[1], [8]], and [[9]] for earlier exact analysis). We assume that each server can work on only one job at a time and that if both servers are assigned to the same station, they can work together on one job. [Extracted from the article]

We study an authentication system that receives requests from different types of users. A centralized controller must assign an authentication method to each request, considering the type, the state of the system and the characteristics of several available methods. Each authentication method has different capacity, service rate, level of security, level of usability and operating cost. We seek to optimize security, usability and operating cost, simultaneously by assigning authentication methods dynamically, in real time. To do this, we model the system as a network of parallel multi-server queues, where each queue represents an authentication method and each customer represents a request. We use two different approaches to handle the multiple objectives: a weighted total cost function, and treating security and latency as constraints while minimizing operating cost. We employ constrained and unconstrained Markov decision processes to determine the structure of policies that effectively balance these three objectives. We conclude that if there are infinitely many servers for each authentication method, then the optimal policy is static. We also show that if one method has finite capacity, then the optimal policy is of trunk reservation form. Our results regarding the structure of the optimal policy are consistent for both modeling approaches. Our work shows that optimal policies have intuitive, easy-to-implement structures that are useful in practice. Under certain assumptions, we provide a straightforward way to obtain an optimal policy. We also offer strategies to use our models to explore non-dominated solutions over the three objective functions. [ABSTRACT FROM AUTHOR]

We study optimal pricing for tandem queueing systems with finite buffers. The service provider dynamically quotes prices to incoming price sensitive customers to maximize the long‐run average revenue. We present a Markov decision process model for the optimization problem. For systems with two stations, general‐sized buffers, and two or more prices, we describe the structure of the optimal dynamic pricing policy and develop tailored policy iteration algorithms to find an optimal pricing policy. For systems with two stations but no intermediate buffer, we characterize conditions under which quoting either a high or a low price to all customers is optimal and provide an easy‐to‐implement algorithm to solve the problem. Numerical experiments are conducted to compare the developed algorithms with the regular policy iteration algorithm. The work also discusses possible extensions of the obtained results to both three‐station systems and two‐station systems with price and congestion sensitive customers using numerical analysis. [ABSTRACT FROM AUTHOR]

We consider optimal pricing for a two-station tandem queueing system with finite buffers, communication blocking, and price-sensitive customers whose arrivals form a homogeneous Poisson process. The service provider quotes prices to incoming customers using either a static or dynamic pricing scheme. There may also be a holding cost for each customer in the system. The objective is to maximize either the discounted profit over an infinite planning horizon or the long-run average profit of the provider. We show that there exists an optimal dynamic policy that exhibits a monotone structure, in which the quoted price is non-decreasing in the queue length at either station and is non-increasing if a customer moves from station 1 to 2, for both the discounted and long-run average problems under certain conditions on the holding costs. We then focus on the long-run average problem and show that the optimal static policy performs as well as the optimal dynamic policy when the buffer size at station 1 becomes large, there are no holding costs, and the arrival rate is either small or large. We learn from numerical results that for systems with small arrival rates and no holding cost, the optimal static policy produces a gain quite close to the optimal gain even when the buffer at station 1 is small. On the other hand, for systems with arrival rates that are not small, there are cases where the optimal dynamic policy performs much better than the optimal static policy. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the dynamic server allocation problem for tandem queueing systems with an equal number of stations and servers. The servers are flexible, yet non-collaborative, so that at most one server can work at a station at any time. The objective is to maximize the long-run average throughput. We show that if each server is the fastest at one station, then a dedicated server assignment policy is optimal for systems of arbitrary size and with general service requirement distributions. Otherwise, the optimal policy is more complex as servers must divide their time between stations. For Markovian systems with two stations and two servers, we characterize the optimal policy completely. For larger Markovian systems, we use our results for two-station systems to propose four heuristic server assignment policies and provide computational results that show that our heuristics are near-optimal. We also compare collaborative and non-collaborative settings to evaluate the benefits of dynamic server allocation, as opposed to collaboration, in systems with flexible servers. We conclude that the loss in the long-run average throughput due to lack of collaboration is mitigated by the similarity of the tasks in the system, and cross-training can still be beneficial in non-collaborative systems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A fundamental aspect of designing systems with dedicated servers is identifying and improving the system bottlenecks. We extend the concept of a bottleneck to networks with heterogeneous, flexible servers. In contrast with a network with dedicated servers, the bottlenecks are not a priori obvious, but can be determined by solving a number of linear programming problems. Unlike the dedicated server case, we find that a bottleneck may span several nodes in the network. We then identify some characteristics of desirable flexibility structures. In particular, the chosen flexibility structure should not only achieve the maximal possible capacity (corresponding to full server flexibility), but should also have the feature that the entire network is the (unique) system bottleneck. The reason is that it is then possible to shift capacity between arbitrary nodes in the network, allowing the network to cope with demand fluctuations. Finally, we specify when certain flexibility structures (in particular chaining, targeted flexibility, and the 'N' and 'W' structures from the call center literature) possess these desirable characteristics. [ABSTRACT FROM AUTHOR]

We study the dynamic assignment of cross-trained servers to stations in understaffed lines with finite buffers. Our objective is to maximize the production rate. We identify optimal server assignment policies for systems with three stations, two servers, different flexibility structures, and either deterministic service times and arbitrary buffers or exponential service times and small buffers. We use these policies to develop server assignment heuristics for Markovian systems with larger buffer sizes that appear to yield near-optimal throughput. In the deterministic setting, we prove that the best possible production rate with full server flexibility and infinite buffers can be attained with partial flexibility and zero buffers, and we identify the critical skills required to achieve this goal. We then present numerical results showing that these critical skills, employed with an effective server assignment policy, also yield near-optimal throughput in the Markovian setting, even for small buffer sizes. Thus, our results suggest that partial flexibility is sufficient for near-optimal performance, and that flexibility structures that are effective for deterministic and infinite-buffered systems are also likely to perform well for finite-buffered stochastic systems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider tandem lines with finite buffers and flexible, heterogeneous servers that are synergistic in that they work more effectively in teams than on their own. Our objective is to determine how the servers should be assigned dynamically to tasks in order to maximize the long-run average throughput. In particular, we investigate when it is better to take advantage of synergy among servers, rather than exploiting the servers' special skills, to achieve the best possible system throughput. We show that when there is no trade-off between server synergy and servers' special skills (because the servers are generalists who are equally skilled at all tasks), the optimal policy has servers working in teams of two or more at all times. Moreover, for Markovian systems with two stations and two servers, we provide a complete characterization of the optimal policy and show that, depending on how well the servers work together, the optimal policy either takes full advantage of servers' special skills, or full advantage of server synergy (and hence there is no middle ground in this case). Finally, for a class of larger Markovian systems, we provide sufficient conditions that guarantee that the optimal policy should take full advantage of server synergy at all times. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The article discusses the study on the assignment of flexible servers in throughput maximization in non-Markovian tandem lines with finite buffers. It states that the study provides theoretical and numerical results and considers various objectives to determine optimal server policy validity. Results verify the robustness of optimal server assignment policy to the service time distributions. It proposes the server assignment heuristics for systems with various service time distributions.

The article discusses the optimal static prices for a finite capacity queueing system that serves customers from various classes. It presents that the multi-class formulation can be reformulated as a single dimensional problem wherein the total load is the decision variable. This alternative formulation prove that the upper bound for optimal arrival rates for queueing systems and offer conditions that ensure a unique optimal arrival rate vector.

We study a service facility modeled as a queueing system with finite or infinite capacity. Arriving customers enter if there is room in the facility and if they are willing to pay the price posted by the service provider. Customers belong to one of a finite number of classes that have different willingnesses-to-pay. Moreover, there is a penalty for congestion in the facility in the form of state-dependent holding costs. The service provider may advertise class-specific prices that may fluctuate over time. We show the existence of a unique optimal stationary pricing policy in a continuous and unbounded action space that maximizes the long-run average profit per unit time. We determine an expression for this policy under certain conditions. We also analyze the structure and the properties of this policy. [ABSTRACT FROM AUTHOR]

For a general class of closed feed-forward fork and join queueing networks, we derive the tail asymptotics of transient cycle times and waiting times under the assumption that the service time distribution of at least one station is subexponential. We also argue that under certain conditions, the asymptotic tail distributions remain the same for stationary cycle times and waiting times. [ABSTRACT FROM AUTHOR]

We consider the problem of maximizing capacity in a queueing network with flexible servers, where the classes and servers are subject to failure. We assume that the interarrival and service times are independent and identically distributed, that routing is probabilistic, and that the failure state of the system can be described by a Markov process that is independent of the other system dynamics. We find that the maximal capacity is tightly bounded by the solution of a linear programming problem and that the solution of this problem can be used to construct timed, generalized round-robin policies that approach the maximal capacity arbitrarily closely. We then give a series of structural results for our policies, including identifying when server flexibility can completely compensate for failures and when the implementation of our policies can be simplified. We conclude with a numerical example that illustrates some of the developed insights. [ABSTRACT FROM AUTHOR]

In a G/GI/c loss system with balking, reneging, or limited waiting space, deleting some of the arriving customers can either increase or decrease the fraction of the remaining arrivals who get served, depending on how customers are deleted. We present a model in which the random deletion of arrivals independently and with some fixed probability can never decrease the fraction of the remaining arrivals who get served. [ABSTRACT FROM AUTHOR]

For a Markovian queueing network with two stations in tandem, finite intermediate buffer, and M flexible servers, we study how the servers should be assigned dynamically to stations to obtain optimal long-run average throughput. We assume that each server can work on only one job at a time, that several servers can work together on a single job, and that the travel times between stations are negligible. Under these assumptions, we completely characterize the optimal policy for systems with three servers. We also provide a conjecture for the structure of the optimal policy for systems with four or more servers that is supported by extensive numerical evidence. Finally, we develop heuristic server-assignment policies for systems with three or more servers that are easy to implement, robust with respect to the server capabilities, and generally appear to yield near-optimal long-run average throughput. [ABSTRACT FROM AUTHOR]

This note discusses the relationships among three assumptions that appear frequently in the pricing/revenue management literature. These assumptions are mostly needed for analytical tractability, and they have the common property of ensuring a well-behaved "revenue function." The three assumptions are decreasing marginal revenue with respect to demand, decreasing marginal revenue with respect to price, and increasing price elasticity of demand. We provide proofs and examples to show that none of these conditions implies any other. However, they can be ordered from strongest to weakest over restricted regions, and the ordering depends upon the region. [ABSTRACT FROM AUTHOR]

This paper is concerned with the design of dynamic server assignment policies that maximize the capacity of queueing networks with flexible servers. Flexibility here means that each server may be capable of performing service at several different classes. We assume that the interarrival times and the service times are independent and identically distributed, and that routing is probabilistic. We also allow for server switching times, which we assume to be independent and identically distributed. We deduce the value of a tight upper bound on the achievable capacity by equating the capacity of the queueing network model with that of a limiting deterministic fluid model. The maximal capacity of the deterministic model is given by the solution to a linear programming problem that also provides optimal allocations of servers to classes. We construct particular server assignment policies, called generalized round-robin policies, that guarantee that the capacity of the queueing network will be arbitrarily close to the computed upper bound. The performance of such policies is studied using numerical examples. [ABSTRACT FROM AUTHOR]

Presents a study that examined a finite-capacity queueing system to analyze nonstationary admission control problems in which the system capacity, and the arrival and service rates, may change over time. Background on bias optimal admission control policies; Conditions for a bias optimal policy to be monotonic in time; Results and discussion.

Expediting defined as using overtime or subcontracting to supplement regular production. This is usually done when the number of backorders has grown to be unacceptably large. In this paper, we consider analytic models for deciding when and how to expedite in a single-product make-to-order environment. We derive the structure of optimal expediting policy in both continuous and discrete-time cases. The continuous-time model corresponds best to subcontracting and the discrete-time model corresponds to either overtime or subcontracting. Models for performance analysis of the continuous-time case are also given. [ABSTRACT FROM AUTHOR]

For a system of finite queues, we study how servers should be assigned dynamically to stations in order to obtain optimal (or near-optimal) long-run average throughput. We assume that travel times between different service facilities are negligible, that each server can work on only one job at a time, and that several servers can work together on one job. We show that when the service rates depend only on either the server or the station (and not both), then all nonidling server assignment policies are optimal. Moreover, for a Markovian system with two stations in tandem and two servers, we show that the optimal policy assigns one server to each station unless that station is blocked or starved (in which case the server helps at the other station), and we specify the criterion used for assigning servers to stations. Finally, we propose a simple server assignment policy for tandem systems in which the number of stations equals the number of servers, and we present numerical results that show that our policy appears to yield near-optimal throughput under general conditions. [ABSTRACT FROM AUTHOR]

(Max,+) linear systems can be used to represent stochastic Petri nets belonging to the class of event graphs. This class contains various instances of queueing networks like acyclic or cyclic fork-and-join queueing networks, finite or infinite capacity tandem queueing networks with various types of blocking, synchronized queueing networks and so on. It also contains some basic manufacturing models such as kanban networks, assembly systems and so forth. In their 1997 paper, Baccelli, Hasenfuss and Schmidt provide explicit expressions for the expected value of the waiting time of the nth customer in a given subarea of a (max,+) linear system. Using similar analysis, we present explicit expressions for the moments and the Laplace transform of transient waiting times in Poisson driven (max,+) linear systems. Furthermore, starting with these closed form expressions, we also derive explicit expressions for the moments and the Laplace transform of stationary waiting times in a class of (max,+) linear systems with deterministic service times. Examples pertaining to queueing theory are given to illustrate the results. [ABSTRACT FROM AUTHOR]

We give a Taylor series expansion for the joint Laplace transform of stationary waiting times in open (max,+)-linear stochastic systems with Poisson input. Probabilistic expressions are derived for coefficients of all orders. Even though the computation of these coefficients can be hard for certain systems, it is sufficient to compute only a few coefficients to obtain good approximations (especially under the assumption of light traffic). Combining this new result with the earlier expansion formula for the mean stationary waiting times, we also provide a Taylor series expansion for the covariance of stationary waiting times in such systems. It is well known that (max,+)-linear systems can be used to represent stochastic Petri nets belonging to the class of event graphs. This class contains various instances of queueing networks like acyclic or cyclic fork-and-join queueing networks, finite or infinite capacity tandem queueing networks with various types of blocking, synchronized queueing networks and so on. It also contains some basic manufacturing models such as kanban networks, assembly systems and so forth. The applicability of this expansion technique is discussed for several systems of this type. [ABSTRACT FROM AUTHOR]

Examines a method for computing tight numerical bounds on renewal functions. Use of Riemann-Stieltjes integration; Provision of bounds for solving renewal equations; Error analysis for numerical bounds; Accuracy of the computed numerical bounds.