Optimal Control of Flexible Servers in Two Tandem Queues With Operating Costs

Abstract

We consider two-stage tandem queuing systems with dedicated servers in each station and flexible servers that can serve in both stations. We assume exponential service times, linear holding costs, and operating costs incurred by the servers at rates proportional to their speeds. Under conditions that ensure the optimality of nonidling policies, we show that the optimal allocation of flexible servers is determined by a transition-monotone policy. Moreover, we present conditions under which the optimal policy can be explicitly determined. Andradottir, Ayhan,, Gel, Hopp, and Van Oyen [10,11], Hopp, Tekin, and Van Oyen [12], Iravani, Van Oyen, and Sims [15], and Ahn and Righter [3]. These models include collaborative and noncollaborative work disciplines, open systems with external arrivals that are independent of the system state, closed systems (e.g., CONWIP systems where a departure triggers a new arrival), and various forms of cross-training (e.g., full, zone, or hierarchical cross-training).There is also much research on the optimal use of flexible servers in tandem systems with holding costs. Because the mathematical models involved are quite complex, most results refer to two-stage systems and exponential service times. Rosberg, Varaiya, and Walrand [19] considered a system with Poisson arrivals, a server with a constant service rate in the downstream station, and a server with a controllable service rate in the upstream station. They showed that the optimal service rate is nondecreasing in the length of the first queue and nonincreasing in the length of the second queue. Weber and Stidham [23] considered a system of n stations with arrivals at each station, where, in addition to convex holding costs, servers incur operating costs that are convex functions of their service rates. They showed that the optimal policy is transition-monotone; that is, when a job leaves a queue, the optimal service rate in that queue is not increased and the optimal service rates in the other queues are not decreased. Duenyas, Gupta, and Olsen [7] and Iravani, Posner, and Buzacott [14] characterized optimal policies for models of n and two-stage systems, respectively, with one flexible server and setup costs. Pandelis and Teneketzis [18] studied two-stage clearing systems with multiple flexible servers where jobs join the second queue with probability p after completing service in the upstream station. They derived conditions under which the policy that gives priority to jobs in the upstream station is optimal for general service times. For a two-stage clearing system with two flexible servers, Ahn, Duenyas, and Zhang [2] provided necessary and sufficient conditions under which an exhaustive policy for the upstream or the downstream station is optimal. Similar results were obtained by Ahn, Duenyas, and Lewis [1] for the model with arrivals. The results of Ahn et al. [2] have been extended in two more directions. First, Schiefermayr and Weichbold [20] obtained the optimal policy for all values of holding costs and service times. Se...

Workforce flexibility in operations management

“…A small portion of the workforce is set to be available to float as generalists. The use of floaters potentially creates additional costs such as higher worker wages, set-up costs, set-up times, and longer working times (due to less task-specific experience) [45,46]. The level of flexibility increases linearly with increased portion of floaters in the system, yet costs associated with the use of floaters grow very fast, and can quickly outweigh the benefits.…”

“…The use of floaters in clearing systems (no external arrival of jobs until originally scheduled jobs clear) has been well studied [46][47][48][49]. Particularly, dynamic allocation of floaters in clearing systems of two tandem queues have been modeled as DP problems.…”

“…Pandelis [49] extended these models and considered conditions of jobs leaving the system after leaving the first station, as well as less constrained cases where the floater can work on both stations. Pandelis [46] later considered the operating cost of using floaters, finding that it may be optimal to idle the floater when the operating costs are higher than holding costs.…”

Optimal control of noncollaborative servers in two‐stage tandem queueing systems

“…Wu et al [25] showed the optimality of a switching-curve policy for the previous model with arrivals and no dedicated server in the upstream station and Pandelis [18] extended this result to the case when jobs may not require service at the downstream station and processing requirements are not the same in each station. Finally, Pandelis [17] studied a model with server operating costs in addition to holding costs and identified conditions under which the switching-curve structure of the optimal policy is preserved.…”

Markov decision processes with multidimensional action spaces

“…First, we include in the model costs that not only depend on the state of the process but on the actor-action pair as well. Such a system is the two-stage queueing model with multiple collaborating servers considered in Pandelis (2008a), where each server incurs operating costs at rates depending on the queue where he is allocated. We show that the same action is optimal for all actors if the operating costs rates are proportional to the actors' speeds, with the proportionality constants being functions only of the actions.…”