Approximate mean waiting time in a GI/D/1 queue with autocorrelated times to failures

In this paper, we study process completion time and propose an accurate approximation for the mean waiting time in queues with servers experiencing autocorrelated times to failure, which include only busy periods from a repair completion until the next failure. To do this, we employ a three-parameter renewal approximation that represents the stream of autocorrelated times to failure. The approximation gives rise to a renewal interruption process with two-state Hyper-exponential (H2) times to failure. Then we compute the mean waiting time exactly in a queue experiencing H2 times to failure when the job arrival process is Poisson. This model provides an approximation for the mean waiting time of the original queue having an autocorrelated disruption process. We also propose an accurate approximation for queues with renewal job arrival processes when the server interruption process is general.     

Location and allocation of service units on a congested network

We consider the problem of locating facilities and allocating servers on a congested network (LASCN). Demands for service that originate from the nodes are assumed to be Poisson distributed and the servers provide a service time that is exponentially distributed. The objective is to minimize the total cost of the system which includes a fixed installation cost, a variable server cost, a cost associated with travel time and a cost associated with the waiting time in the facility for all the customers. The problem is formulated using non-linear programming and subsequently analyzed. Results for exact and approximate solution approaches are reported. We also show that we can modify the approaches to solve the LASCN with constraints limiting both the travel time to the servers and the waiting time of customers.     

Optimal Design of Multiserver Queueing Systems with Different Waiting Costs for Time in Queue and Time in Service

The optimal number of servers and service rate are characterized for multiserver systems in which waiting times in service and in queue have different costs. It is shown that the optimal choice of the service rate and the number of servers depends on the ratio of the in-service waiting cost coefficient to the in-queue waiting cost coefficient. For M/M/c systems, a single server is optimal if the cost-coefficient ratio exceeds a threshold value of one-half. For GI/M/c and M/G/c systems, the threshold ratio increases as the variation of the interarrival or service time increases.     

Optimal control of servers in front and back rooms with correlated work

In this article we deal with the problem of optimal server allocation between the front and the back room of a service facility (for all states of the system) when the work in the back room is generated by the service provided in the front room. The servers are cross-trained and can switch between front room and back room tasks without any loss of productivity. We also determine the optimal number of servers required to meet certain service levels. Our service criteria is that both the mean number of customers waiting for service in the front room and the mean number of jobs (work) accumulated are bounded from above. We model the problem as a Markov decision problem and use linear programming to solve it.     

Blocking Probabilities for a Class of Two-Station Tandem Queueing Models

We study a two-station tandem queueing system with a finite buffer of maximum size M (M≥0) between the stations. The first service station has L≥1(homogeneous) parallel servers with an unlimited waiting space, while at the second there are N≥1 (homogeneous) parallel servers. Assuming Poisson arrivals and exponential service times, we derive exact results for the blocking probabilities for some special cases, and we present a unified approach for the analysis of this class of models under saturation conditions. Practical applications of the model are also discussed.     

A CAPACITY PLANNING MODEL FOR A MULTI-PRODUCT FACILITY

A deterministic capacity planning model for a multi-product facility is analyzed to determine (he sizes to be expanded (or disposed of) in each period so as to supply the known demand for N products on time and to minimize the total cost incurred over a finite planning horizon of T periods. The model assumes that each capacity unit of the facility simultaneously serves a prespecified number of demand units of each product, that costs considered include capacity expansion costs, capacity disposal costs, and excess (idle) capacity holding costs, and all the associated cost functions are nondecreasing and concave, and that backlogging is not allowed. The structure of an optimal solution is characterized and then used in developing an efficient dynamic programming algorithm that finds optimal capacity planning policies. The required computational effort is a polynomial function of N and T.     

On the transient behaviour of a model for queues in series with finite capacity

Numerical studies have been carried out on the transient behaviour of a model for queues in series with finite waiting space. Poisson input and exponential service times have been assumed. On the basis of numerical results various conclusions have been drawn both for the transient and steady state situations concerning the servers arrangements, overflow probability, utilization etc     

On a Multichannel Queueing System with Ordered Entry and Heterogeneous Servers

A multichannel queueing system with ordered entry and no waiting line is considered under the assumption of Poisson arrival and negative exponential service with a different rate in each channel. For the system with two or three heterogeneous servers, the best servers' arrangement is established by interchanging the order of servers to decrease the overflow probability and thereby increase the number of items serviced per unit time. The system considered is a simplified model for a production system with a closed loop conveyor.     

DETERMINISTIC APPROXIMATIONS FOR INVENTORY MANAGEMENT AT SERVICE FACILITIES

We consider the problem of finding optimal inventory policies for a service facility where the demand for inventory occurs during the provision of service (e.g., fixing a car in a repair shop). The paper formulates a model where both the demand and service rates are assumed to be constant and deterministic. Consequently a queue forms only during stockouts. In the first of two models analyzed, the service rate is assumed to be fixed and cannot be controlled by the service facility. The ability to use this simple, deterministic model to approximate systems with probabilistic arrival and/or service distributions is also analyzed. The second model relaxes the assumption of a fixed service rate. Optimal inventory policies are derived under linear costs over ordering, inventory holding, customer waiting, and the service rate.     

Repair policies for additively degrading machines

This paper describes a method for determining repair policies for machines whose output degrades additively over time. The novelty of the models is that they consider both the state of the machines as well as the state of the repair facility when making repair decisions. The objective in the models is to maximize long-run production. In one model, we approximate the queue waiting time by a geometric random variable, while in the second model we approximate the waiting time by a sequence of geometric random variables (with different means). We show that as the average repair queue increases, the decision to repair must be made earlier. In addition, we show empirically that the simple geometric waiting time approximation becomes less accurate as the queue length increases and that the approximation understates the expected long-run output of the machine. A plastic moulding facility is used to motivate the problem. Computational results using industry supplied data are presented. The results indicate that substantial (10-20%) productivity improvement can be realized using the derived repair policies instead of policies that do not consider the repair queue.     

Detection of an appropriate pharmaceutical company to get a suitable vaccine against COVID-19 with minimum cost under the quality control process

Some international pharmaceutical companies have succeeded in producing vaccines against COVID-19. Countries all over the world have aimed to obtain these vaccines with minimum cost. We consider a set of K-independent Markovian waiting lists. Each list contains a set of countries, where each one of them has an exponential service time and a Poisson arrival process. These companies differ in some characteristics such as the vaccine production cost and the speed of the required quantity delivery. We present a new detection model that helps in providing an appropriate decision to choose a suitable company. Moreover, the concept of balking and the retention of reneged countries is taken into consideration under the quality control process of each waiting list. Under steady state, we face an interesting and difficult discrete stochastic optimization problem. Its solution gives an optimal distribution of the searching effort, which is bounded by a known probability distribution. A simulation study has been derived to get the minimum value of the paid cost random values. The highest service rate, the total expected profit of each queuing system, and the optimum performance measures, which depend on this cost, have been obtained to show the effectiveness of this model.     

A periodic-review inventory model in a fluctuating environment

We study a single-item periodic-review inventory model in a fluctuating environment with a fixed lead time of λ periods. The state of the environment at the beginning of each period is described by a homogeneous Markov chain. Ordering, holding, penalty costs and the distributions of random variables representing the customer's demand and the supplier's capacity level are state environment dependent. By using dynamic programming it is proved in the finite-horizon case that the optimal policy is of a base stock type. Its parameters are monotonic in the number of the periods making up the horizon and also in stochastically ordered random variables representing different supplier capacities. Similar results are proved for the infinite-horizon problem.     

An optimal path of a moving vehicle on a sphere

We consider the problem of finding an optimal path of a moving vehicle, which during its journey provides a certain type of service to a group of existing facilities on a sphere. Service requests from each facility are assumed to follow a Poisson process and the conditional expected cost incurred by a service call is assumed to be proportional to the shortest arc distance between the facility and the vehicle's location at the time of the service request. This problem is formulated as a variational problem and solved by repeatedly applying a dynamic programming procedure to a sequence of staged networks that are constructed via perturbation.     

Replenishment planning in discrete-time, capacitated, non-stationary, stochastic inventory systems

In this paper, we examine a single-product, discrete-time, non-stationary, inventory replenishment problem with both supply and demand uncertainty, capacity limits on replenishment quantities, and service level requirements. A scenario-based stochastic program for the static, finite-horizon problem is presented to determine replenishment orders over the horizon. We propose a heuristic that is based on the first two moments of the random variables and a normal approximation, whose solution is compared with the optimal from a simulation-based optimization method. Computational experiments show that the heuristic performs very well (within 0.25% of optimal, on average) even when the uncertainty is non-normal or when there are periods without any supply. We also present insights obtained from sensitivity analyses on the effects of supply parameters, shortage penalty costs, capacity limits, and demand variance. A rolling-horizon implementation is illustrated.     

A Sequential Bounding Approach for Optimal Appointment Scheduling

This study is concerned with the determination of optimal appointment times for a sequence of jobs with uncertain durations. Such appointment systems are used in many customer service applications to increase the utilization of resources, match workload to available capacity, and smooth the flow of customers. We show that the problem can be expressed as a two-stage stochastic linear program that includes the expected cost of customer waiting, server idling, and a cost of tardiness with respect to a chosen session length. We exploit the problem structure to derive upper bounds that are independent of job duration distribution type. These upper bounds are used in a variation of the standard L-shaped algorithm to obtain optimal solutions via successively finer partitions of the support of job durations. We present new analytical insights into the problem as well as a series of numerical experiments that illustrate properties of the optimal solution with respect to distribution type, cost structure, and number of jobs.     

Make-to-order versus make-to-stock in a production–inventory system with general production times

We study the optimality of make-to-order (MTO) versus make-to-stock (MTS) policies for a company producing multiple heterogeneous products at a shared manufacturing facility. Manufacturing times are general i.i.d. random variables, and different products may have different manufacturing-time distributions. Demands for the products are independent Poisson processes with different arrival rates. The costs of managing the production-inventory system are stationary and include inventory holding and backordering costs. Backordering costs may be $ per unit or $ per unit per unit time. We derive optimality conditions for MTO and MTS policies. We also study the impact of several managerial considerations on the MTO versus MTS decision.     

Reducing cycle times in manufacturing and supply chains by input and service rate smoothing

A key performance parameter of a manufacturing network or supply chain is its cycle time; the time that a typical item spends in the network. A previous simulation study on a semiconductor assembly and test facility showed that cycle times could be reduced by having smooth input and service rates. This suggested that there is a “cycle time principle” that, for a system with a specified throughput or input rate, the shortest cycle times are obtained when the input and service rates do not vary over time. We prove that this principle is true for the M/G/1 and M/M/s queueing systems and Jackson networks. The analysis involves establishing several results on the concavity of waiting time probabilities and the convexity of expected waiting times and queue lengths, as functions of input and service rates. These results also have natural uses in other optimization problems.     

Approximation of the probability distribution of the customer waiting time under an (r,s, q) inventory policy in discrete time

We study a single-item periodic review (r, s, q) inventory policy. Customer demands arrive on a discrete (e.g. daily) time axis. The replenishment lead times are discrete random variables. This is the time model underlying the majority of the Advanced Planning Software systems used for supply chain management in industrial practice. We present an approximation of the probability distribution of the customer waiting time, which is a customer-oriented performance criterion that captures supplier–customer relationships of adjacent nodes in supply chains. The quality of the approximation is demonstrated with the help of a simulation experiment.     

Finite waiting space bulk service system

Summary This paper discusses the ergodic queue length distribution of a bulk service system with finite waiting space by the method of the imbedded Markov chain. The system under consideration is a queuing system with Poisson arrivals, general service times, single server and where service is performed on batches of random size.     

Effects of Slow-Downs and Failure on Stochastic Service Systems

The reliability and efficiency of a stochastic service system subject to break-downs and slow-downs is investigated. A mathematical model of such a system is a single server queue with Poisson input and exponential service time. However, the server is subject to break-downs and slow-downs, the lengths of which are exponentially distributed. During these periods he serves at varying rates. Analytic expressions are derived for the mean waiting time and the mean queue length.