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.Markov Decision Processes, Markovian Queueing Systems, Tandem Queues, Finite Buffers, Mobile and Cooperating Servers, Preemptive Service, Manufacturing Blocking

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 in the network. 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.

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 in order 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.

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.

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.

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We consider a finite capacity queueing system in which each arriving customer offers a reward. A gatekeeper decides based on the reward offered and the space remaining whether each arriving customer should be accepted or rejected. The gatekeeper only receives the offered reward if the customer is accepted. A traditional objective function is to maximize the gain, that is, the long-run average reward. It is quite possible, however, to have several different gain optimal policies that behave quite differently. Bias and Blackwell optimality are more refined objective functions that can distinguish among multiple stationary, deterministic gain optimal policies. This paper focuses on describing the structure of stationary, deterministic, optimal policies and extending this optimality to distinguish between multiple gain optimal policies. We show that these policies are of trunk reservation form and must occur consecutively. We then prove that we can distinguish among these gain optimal policies using the bias or transient reward and extend to Blackwell optimality.

Consider a system of queuing stations in tandem having both flexible servers (who are capable of working at multiple stations) and dedicated servers (who can only work at the station to which they are dedicated). We study the dynamic assignment of servers to stations in such systems with the goal of maximizing the long-run average throughput. We also investigate how the number of flexible servers influences the throughput and compare the improvement that is obtained by cross-training another server (i.e., increasing flexibility) with the improvement obtained by adding a resource (i.e., a new server or a buffer space). Finally, we show that having only one flexible server is sufficient for achieving near-optimal throughput in certain systems with moderate to large buffer sizes (the optimal throughput is attained by having all servers flexible). Our focus is on systems with generalist servers who are equally skilled at all tasks, but we also consider systems with arbitrary service rates.Ayhan, and Down [6][7][8], and Tassiulas and Bhattacharya [29] considered the dynamic assignment of servers to maximize the long-run average throughput in queuing networks with flexible servers. However, these articles do not focus on the case where some servers are dedicated to specific stations.Ostalaza, McClain, and Thomas [23], McClain, Thomas, and Sox [22], Zavadlav, McClain, and Thomas [34], and, more recently, Ahn and Righter [4] have considered using server flexibility to achieve dynamic line balancing. More specifically, Ostalaza et al. [23] and McClain et al. [22] studied dynamic line balancing in tandem queues with shared tasks that can be performed at either of two successive stations. This work was continued by Zavadlav et al. [34], who studied several server assignment policies for systems with fewer servers than stations, in which all servers trained to work at a particular station have the same capabilities at that station. Ahn and Righter [4] studied how workers who are trained to do a set of consecutive tasks should be assigned dynamically to tandem stations. Bartholdi and Eisenstein [9] defined the "bucket brigades" server assignment policy for systems in which each server works at the same rate at all tasks and showed that under this policy, a stable partition of work will emerge yielding optimal throughput. Bartholdi, Eisenstein, and Foley [10] showed that the behavior of the bucket brigades policy, applied to systems with discrete tasks and exponentially distributed task times, resembles that of the same policy applied in the deterministic setting with infinitely divisible jobs.Gurumurthi and Benjaafar [14], Hopp, Tekin, and van Oyen [17], and Sheikhzadeh, Benjaafar, and Gupta [27] considered the use of specific flexibility structures on a set of existing servers to enhance the system's performance (see also Jordan and Graves [19] for related work). More specifically, Gurumurthi and Benjaafar [14] considered the modeling and analysis of flexible queuing systems. They illustrated that for systems with identi...