Optimal Service Rate Research Articles

Quality-Speed Trade-Offs in Dynamic Service Systems: Do Future Demand Changes Matter?

A quality-speed trade-off occurs in many service systems in which employees have discretion over their service times. An increase in service time often increases the quality of the service but also increases congestion. We investigate this trade-off between quality and speed when the service rate can change over time based on exogenous future demand changes. This work is the first analysis of the quality-speed trade-off with exogenous time-dependent demand. It shows how service providers can benefit from optimizing service rates based on future demand changes. We derive structural optimality results from an analysis of stationary stochastic systems. For time-dependent demand, a time-dependent performance approximation based on a stationary backlog-carryover approach is integrated into an optimization model. In addition, we present a smoothing heuristic that employs the stationary results. The impacts of different service value functions and congestion measures are investigated. For waiting costs, which are based on the expected number of customers in the system, the optimal stationary service rates increase in the demand. Counterintuitively, for waiting costs, which are based on the expected sojourn time, the optimal stationary service rates first show a decrease and then an increase in the demand. For time-dependent service rates, the integrated model delivers reliable results. Even though such service systems cannot produce to stock, future demand changes influence time-dependent service rates several periods beforehand. Substantial benefits compared to stationary approaches can be achieved by considering time-dependent demand in the decision on service rates, especially for cases with a low waiting cost and a moderate to high sensitivity of quality to the service rate.

Machine Repair System with F-Policy, Two Unreliable Servers, and Warm Standbys

Abstract This investigation deals with machine repair problems with F-policy and a repair facility that has two unreliable servers and the provision of warm standbys. F-policy implies that when the number of failed machines reaches a maximum capacity K of the system, the next arrival of failed machines is not allowed into the system until the number of failed machines again decreases to a fixed level F. The first server repairs the failed machines as soon as they fail, but the second server starts the repair service only when there are more than a prespecified number of failed machines available in the system to minimize the work load of failed machines in the system. Both servers may be broken down during the repair of failed machines and, after repair, again return to the system for service to the failed machines. We developed a Markov model for the machine repair problem, which consists of the combination of operating and standby machines using the birth-death process. The matrix method is employed to determine the transient probabilities of the number of failed machines in the system, throughput of the system, and some other performance measures. The cost function is also provided for the determination of optimal service rates to depict the system utility at optimum cost.

Optimal Service Rates of a perishable Inventory System with Service Facility

This paper studies a continuous-review perishable inventory system with service facility. We consider a finite capacity inventory system with Markovian arrivals and exponentially distributed service times. The inventory is replenished according to an (s, S) policy and the lead times follow phase type distribution. The life time of each item is assumed to be exponential. By controlling the service rate at each instance of time, we can minimise the total expected cost rate. We formulate this model as a semi Markov decision problem. The stationary optimal policy is computed using linear programming algorithm and the results are illustrated numerically.

Dynamic Pricing Control for Open Queueing Networks

Pricing control is an important problem in service systems and it aims to control customer behaviors through an economic way, instead of administrative commands. In this paper, we study a dynamic pricing and service rate control problem in an open Jackson network with limited capacity. The goal is to determine the optimal admission prices and the optimal service rates at every state such that the long-run average social welfare is maximized. The original problem is decomposed into a rate-setting problem plus a price-setting problem. To solve the rate-setting problem, we derive a difference formula based on the sensitivity-based optimization theory. When the cost rate function is convex in service rates and the value rate function is concave in arrival rates, we decompose the rate-setting problem into a series of convex optimization subproblems. When the rate functions have linear structure, these subproblems are even simpler and a bang–bang control is optimal. For the price-setting problem, we determine the state-dependent prices so as to induce the optimal arrival rates obtained by the rate-setting problem. We propose a recursive algorithm to numerically compute the conditional expected delays at every state. Finally, we conduct numerical experiments to explore the optimality properties and some useful insights for this dynamic pricing control problem.

Minimal cost service rate in priority queuing models for emergency cases in hospitals

Performance measures and waiting time cost for higher priority patients with severe cases over lower priority patients with stable cases using preemptive priority queuing model were obtained. Also, a total expected waiting time cost per unit time for service and the expected service cost per unit time for priority queuing models: M/M/2: ∞/NPP and M/M/2: ∞/PP were respectively formulated and optimized to obtain optimum cost service rate that minimizes the total cost. The results were applied to obtain optimum service rate that minimizes the total cost of providing and waiting for service at the emergency consulting unit of hospital.   Â

Managing operations in customer-intensive services with forward-looking customers

PurposeThis paper aims to investigate the optimal price and service rate decisions in a customer-intensive service, where customers’ perceived service quality decreases in the service speed. Customers are assumed to be forward-looking in purchase decision-making and heterogeneous in their reservation utilities. The purpose of this paper is to study the impact of customers’ forward-looking behavior and the heterogeneity on the operational decisions in a customer-intensive context.Design/methodology/approachThe service is delivered through an M/M/1 queue system with unobservable queues. Customers are forward-looking in queue joining decisions, where the purchase decisions are made when the expected utility is greater than the reservation utility. The optimal price and service rate decisions are analyzed with both homogeneous and heterogeneous customers, where homogenous customers have the same reservation utility in purchase decision-making, while heterogeneous customers have different reservation utilities, which are captured by a random variable.FindingsThe optimal price and service rate decisions with forward-looking customers depend on the customer intensity, potential market size and customers’ reservation utility distribution. The results suggest that customers’ heterogeneity in terms of their reservation utilities affects the optimal decisions, market coverage and the expected revenue. Service providers need to take customers’ heterogeneity and the forward-looking behavior into operational decision-making.Originality/valueThis paper extends previous studies in customer-intensive service and contribute to the service operations management area by explicitly incorporating customers’ forward-looking behavior and heterogeneity in purchase decision-making. Assuming customers are forward-looking and heterogeneous is more realistic and practical. The results highlight that knowing customers’ behavioral characteristics can better improve decision-making in service operations, which is critical for enhancing customers’ satisfaction and loyalty, thus critical to a firm’s success in the market with intensive competition.

Performance and cost analysis of a finite capacity queue with two heterogeneous servers under F-policy

All previous works on F-policy queues with multiple servers have focused on multiple homogeneous servers. This paper presents an analytical analysis on the F-policy M/M/2/K queue with heterogeneous servers. We obtained the steady-state distribution of the number of customers in the system and developed a variety of performance measures. The expected cost function was constructed to determine the optimal threshold value and optimal service rates with the objective of minimising the expected cost per unit time. To this end, we developed an algorithm based on the quasi-Newton method to solve the problem of cost minimisation. Finally, sensitivity analysis was conducted using numerical examples.

Optimal Service Rate in Cognitive Radio Networks With Different Queue Length Information

Cognitive radio (CR) technology effectively overcomes spectrum inefficiency by providing the capability that unlicensed users can share the radio frequency spectrum with licensed users. In this paper, we consider a CR system with heterogeneous users. A single primary user (PU) randomly generates service requests and a licensed band processes these requests. This PU band also can be flexibly accessed by secondary users (SUs) if available. The CR system is regarded as a preemptive priority queueing system. Under different information levels, we investigate the equilibrium strategic behaviors of PU and SUs. Based on users’ strategies, we study SU’s sojourn time (i.e., the period beginning from the time an SU request enters the system and ending from the time the SU request is completed), and obtain the analytical solutions of SU’s mean sojourn time. By theoretical and numerical analysis, the SU’s mean sojourn time is found not decreasing with the service rate of PU. This phenomenon is counterintuitive, and it implies the increase of the service rate of PU does not necessarily reduce the mean sojourn time of SU. In this sense, we investigate and find optimal service rates of PU in different information levels to meet the PU’s QoS requirement and simultaneously to maximize SU’s throughput from the viewpoint of the service providers.

Unreliable retrial queue with loss and feedback under threshold-based policy

This paper considers an M/M/1 unreliable retrial queueing system with geometric loss and feedback under the threshold-based policy. After the customer is served completely, he may decide either to leave the system or to join the retrial orbit again for another service. If the server is found busy, the customer may either leave the system or join the retrial orbit. The server may be broken down at any time during the server is operating. When the server is broken down, it cannot be repaired immediately until the number of customers in the system reaches a specified threshold value. This system is modelled by a quasi-birth-and-death process, and some system performance measures are derived. The formulae for computing the rate matrix and stationary probabilities are derived by means of matrix-analytical approach. We obtain the optimal threshold value and the optimal service rate simultaneously to minimise the cost function of the system.

Service Rate Control of Tandem Queues With Power Constraints

In this paper, we study the optimal control of service rates of a tandem queue with power constraints. The service rate of a server is determined by the power allocated to that server. The total power of the system is fixed. The system cost is comprised of two parts, the holding cost reflecting the congestion of queues and the operating cost reflecting the power consumed at servers. The optimization objective is to find the optimal power allocation policy among servers, which can minimize the system average cost. We formulate this problem as a Markov decision process with a constrained action space. Sensitivity-based optimization theory is applied to study this problem. The necessary and sufficient condition of optimal service rates, and the optimality of the vertexes of the feasible domain are derived when the operating cost has a linear or concave form. An iterative algorithm is further developed to find the optimal service rates. This algorithm may work well even when the cost function has a general form. The extension to general tandem queues with many servers is also studied. Finally, we conduct numerical experiments under different parameter settings to demonstrate the main idea of this paper.

Optimal Control of State-Dependent Service Rates in a MAP/M/1 Queue

In this paper, we study the optimal control of service rates in a queueing system with a Markovian arrival process (MAP) and exponential service times. The service rate is allowed to be state dependent, i.e., we can adjust the service rate according to the queue length and the phase of the MAP. The cost function consists of holding cost and operating cost. The goal is to find the optimal service rates that minimize the long-run average total cost. To achieve that, we use the matrix-analytic methods (MAM) together with the sensitivity-based optimization (SBO) theory. A performance difference formula is derived, which can quantify the difference of the long-run average total cost under any two different settings of service rates. Based on the difference formula, we show that the long-run average total cost is monotone in the service rate and the optimal control is a bang–bang control. We also show that, under some mild conditions, the optimal control policy of service rates is of a quasi-threshold-type. By utilizing the MAM theory, we propose a recursive algorithm to compute the value function related quantities. An iterative algorithm to efficiently find the optimal policy, which is similar to policy iteration, is proposed based on the SBO theory. We further study some special cases of the problem, such as the optimality of the threshold-type policy for the M/M/1 queue. Finally, a number of numerical examples are presented to demonstrate the main results and explore the impact of the phase of the MAP on the optimization in the MAP/M/1 queue.

Optimizing the return window for online fashion retailers with closed-loop refurbishment

The strategic positioning of online fashion retailers is defined, in part, by how they handle the complex task of managing returns. Although customers demand lenient policies such as free and late returns, tight control of returned items is mandated by the high costs of re-transportation and product value erosion. We model closed-loop fashion supply chains in order to describe, analyze, and optimize the performance of both forward and backward networks, including a secondary market. The model is based on a queueing system that combines the effect of new products entering the network for the first time with returned products entering the network for the second or nth time. We derive a closed-form expression for optimal service rates at both the test and refurbishment facilities and for the optimal return window. We then analyze the economic effects—on supply chain profit—of return rate, multiple looping of the same item in the supply chain, returned items being sold in a secondary market, and controlled delays in the refurbishment cycle. We apply our model to data obtained from a German fashion online retailer and report the managerial insights derived from this approach. Our results indicate how a company can set its return window strategically so as to maximize her profit as well as how it can decide whether to refurbish merchandise or sell it in the secondary market. In addition, we describe how refurbishment activities can sometimes lead to greater benefits even though the secondary market is usually an attractive opportunity for product returns.

Optimal Control of Admission Prices and Service Rates in Open Queueing Networks

Pricing control is an important problem in service systems. In this paper, we study a dynamic pricing and control problem in an open Jackson network. The goal is to determine the optimal admission prices and the optimal service rates at every state such that the long-run average welfare of the whole system is maximized. We prove that the original problem is equivalent to a rate-setting problem plus a price-setting problem. To solve the rate-setting problem, we derive a difference formula based on the sensitivity-based optimization theory. When the cost rate function is convex in service rates and the value rate function is concave in arrival rates, we decompose the rate-setting problem into a series of convex optimization sub problems. When the rate functions have linear structure, these sub problems are even simpler and a bang-bang control is optimal. For the price-setting problem, we determine the state-dependent prices so as to induce the optimal arrival rates obtained by the rate-setting problem. We propose a recursive algorithm to numerically compute the expected delays at every state. Finally, we conduct numerical experiments to explore properties of the optimal arrival rates, service rates, prices, and expected delays obtained by our approach.

Analysis of a finite-capacity system with working breakdowns and retention of impatient customers

In this paper, we consider a finite-capacity Markovian queueing system with working breakdowns, reneging, and retention of impatient customers, in which the server is subject to breakdowns and repairs while serving a customer. In a working breakdown queue, the server works at a lower service rate rather than stopping service during the breakdown period. An arriving customer finding the server busy must wait in the queue and may leave the system without being served with probability θ or remain in the queue with probability 1−θ within a period of time. We numerically investigate the transient behavior of the queueing model based on the fourth-order Runge–Kutta method. A matrix method is used to compute the steady-state probabilities explicitly. We develop system performance measures for transient and steady states. Numerical examples are given to examine the effects of various system parameters on the system performance measures. A cost optimization problem is formulated to find the optimal service rates, which minimize the expected cost per unit time. Finally, we present an example illustrating the application of the queueing model in an order picking system.

Performance Analysis of a Discrete-Time Queue with Working Breakdowns and Searching for the Optimum Service Rate in Working Breakdown Period

Abstract This paper deals with a discrete-time Geo/Geo/1 queueing system with working breakdowns in which customers arrive at the system in variable input rates according to the states of the server. The server may be subject to breakdowns at random when it is in operation. As soon as the server fails, a repair process immediately begins. During the repair period, the defective server still provides service for the waiting customers at a lower service rate rather than completely stopping service. We analyze the stability condition for the considered system. Using the probability generating function technique, we obtain the probability generating function of the steady-state queue size distribution. Also, various important performance measures are derived explicitly. Furthermore, some numerical results are provided to carry out the sensitivity analysis so as to illustrate the effect of different parameters on the system performance measures. Finally, an operating cost function is formulated to model a computer system and the parabolic method is employed to numerically find the optimum service rate in working breakdown period.

Simultaneous buffer and service rate allocation in open finite queueing networks

ABSTRACTSimultaneous buffer and service rate allocation in open finite queueing networks is a nonlinear mixed-integer programming problem that is -Hard. A queueing network decomposition methodology is coupled with a nonlinear sequential quadratic programming algorithm to compute the simultaneous optimal buffer allocations and service rates via a branch-and-bound scheme for various network topologies. It is shown that the optimization problem is a nonlinear convex programming problem, which assists in the search for local optimal solutions. The material handling or transportation system for transferring the finite customer population between the nodes in the network is also included. Extensive numerical results demonstrate the efficacy of the methodology for series, split, and merge topology networks. Examination of the persistence or absence of the allocation patterns of the service rates and buffers is one of the focal points of this work.

Admission Control Policy of Maintenance for Unreliable Server Machining System with Working Vacation

This investigation is concerned with the performance modeling of machining system operating under the admission control F-policy and server working vacation policy. The repair of failed machines is provided by an unreliable server, who also renders the service with the slower rate rather than completely terminating the service during the vacation period. The failed machines are allowed to enter the system till the system capacity (K) is full; then after failed machines are not allowed to join the system until the system size again decreases to the prespecified threshold level ‘F’. At that instant, the server takes start-up time in order to start allowing the failed machines to enter into the system for the repair job. Numerical method based on successive over-relaxation is applied to obtain the steady-state probabilities and various performance indices including the cost function. The numerical simulation is performed to explore the sensitivity of the system indices with respect to various parameters. Quasi-Newton method and direct search method are used to determine the optimal service rate and threshold parameter.

Computational analysis of multi-server discrete-time queueing system with balking, reneging and synchronous vacations

This paper proposes a discrete-time multi-server queue with multiple synchronous vacations under balking and reneging. Arriving customers decide whether to join the system or balk on the basis of some state-dependent joining/balking probabilities, and renege according to a geometric distribution when servers are busy. The servers take a vacation together if there are no customers in the system at a service completion instant. When the servers are on vacation, an arriving customer activates an impatience timer which is geometrically distributed. The inter-arrival times, service times and vacation times are assumed to be independent and geometrically distributed. We obtain closed-form expressions and develop a computational algorithm for calculating the steady-state probabilities. Specifically, we establish the application of the proposed framework in analyzing a multi-server queueing system with synchronous vacation under balking and reneging. Applications of such models can be found in a wide variety of real-time systems including call centers, computer and communication systems, cloud computing, quality control and maintenance in industrial establishments. We develop a cost model to determine the optimal service rate. Various performance measures and numerical examples are sketched out to demonstrate the impact of the proposed method. Some special cases of the model have also been discussed. Finally, we show that in the limiting case the results converge to the corresponding continuous-time counterparts.

Optimization of an M/M/∞ Queueing System with Free Experience Service

It is a common practice for online service providers to offer free experience service to attract new clients. However, providing experience service requires resources, which may negatively impact current service level and lead to customer turnover. Therefore, providers need to trade off between consequent benefits and costs. We consider a free experience service system where each arriving customer can start his service immediately without waiting, which is a typical situation of many online service platforms. A queueing model with infinite number of servers is employed to study such a service system. The closed forms of the expected numbers of informed and uninformed customers in steady-state are derived by solving nonhomogeneous linear partial differential equations. After that, the expected profit of the service provider is generated and maximized by determining the optimal price and service rates. Our numerical results suggest that with the increase of the market share and the serving cost for the informed customers, the service provider should lay more emphasis on offering the regular service for the informed customers.

Efficient Modeling and Demand Allocation for Differentiated Cloud Virtual-Network as-a Service Offerings

Cloud clients (CCs) of current distributed cloud applications are still not assured of their service quality, in particular, in terms of the experienced latency. Unfortunately, this is mainly attributed to the unpredictability of the communication links among their hosting distributed data centers. To address this problem, this article introduces a novel virtual-network-as-a-service (VNaaS) model to host these applications. In contrast to existing randomly or statically provisioned inter-data centers bandwidth sharing models, the proposed model allows CCs to accurately express their varying network resources needs, demand constraints and tolerance to the cloud latency. In turn, the model maps these requirements to create inter-data centers virtual links hosting each multiple virtual pipes with differentiated service qualities to carry the CC's various traffic flows. To aid the CCs in optimally determining their VNaaS demands, given the budget constraints of their hosted applications, we also develop a novel demand selection scheme based on a two stage-budget allocation mechanism. In the first budgeting stage, the CC calculates an optimal effective service rate for each of its virtual link along with a corresponding link budget and price index. In the second stage, the virtual link budget is distributed to purchase bandwidth for the link's virtual pipes, each with a given service quality and pricing. We then extend the proposed model to allow the CC to enforce any required virtual links’ capacity constraints on the effective service rates resulting from the traffic matrix on the VNaaS. Finally, we develop corresponding differentiated VNaaS pricing and service monitoring mechanisms that can be employed by the cloud service provider (CSP) to regulate the offerings and demands of the distributed cloud services. Performance evaluation results demonstrate the significant improvement in the service quality, the higher utilization of the cloud resources and the increase in the CSP's net profit.