A comparative study of the inventory system with service facility and postponed demands

In this article, we present a continuous review （s,S） inventory system with a service facility consisting of finite buffer （capacity N ） and a single server. The customers arrive according to a Poisson process. The individual customer＇s unit demand is satisfied after a random time of service, which is assumed to be exponential. When the inventory level drops to s＇an order for Q（= S-s） items is placed. The lead time of reorder is assumed to be exponential distribution. An arriving customer, who finds the buffer is full, enters into the pool of infinite size or leaves the system according to a Bernolli trial. At the time of service completion, if the buffer size drops to a preassigned level L （1 〈 L 〈 N） or below and the inventory level is above s, we select the customers from the pool according to two different policy ： in first policy, with probability p （0 〈 p 〈 1） we select the customer from the head of the pool and we place the customer at the end of the buffer; in the second policy, with p （0 〈 p 〈 1） the customer from the pool is transferred to the buffer for immediate service and after completion of his service we provide service to the customer who is in the buffer with probability one. If at a service completion epoch the buffer turns out to be empty, there is at least one customer in the pool and the inventory level is positive, then the one ahead of all waiting in the pool gets transferred to the buffer, and his service starts immediately. The joint probability distribution of the number of customers in the pool, number of customers in the buffer and the inventory level is obtained in the steady-state case. Various stationary system performance measures are computed and total expected cost rate is calculated. A comparative result of two models is illustrate numerically.     

Analysis of a continuous time SM[K]/PH[K]/1/FCFS queue: Age process, sojourn times, and queue lengths

This paper studies a continuous time queueing system with multiple types of customers and a first-come-first-served service discipline.Customers arrive according to a semi-Markov arrival process and the service times of individual types of customers have PH-distributions.A GI/M/1 type Markov process for a generalized age process of batches of customers is constructed.The stationary distribution of the GI/M/1 type Markov process is found explicitly and,consequently,the distributions of the age of the batch in service,the total workload in the system,waiting times,and sojourn times of different batches and different types of customers are obtained.The paper gives the matrix representations of the PH-distributions of waiting times and sojourn times.Some results are obtained for the distributions of queue lengths at departure epochs and at an arbitrary time.These results can be used to analyze not only the queue length,but also the composition of the queue.Computational methods are developed for calculating steady state distributions related to the queue lengths,sojourn times,and waiting times.     

Analysis of a Discrete-Time Geo/G/1 Queue in a Multi-Phase Service Environment with Disasters

This paper considers a discrete-time Geo/G/1 queue in a multi-phase service environment,where the system is subject to disastrous breakdowns, causing all present customers to leave the system simultaneously. At a failure epoch, the server abandons the service and the system undergoes a repair period. After the system is repaired, it jumps to operative phase i with probability qi, i = 1, 2 ···, n.Using the supplementary variable technique, we obtain the distribution for the stationary queue length at the arbitrary epoch, which are then used for the computation of other performance measures. In addition, we derive the expected length of a cycle time, the generating function of the sojourn time of an arbitrary customer, and the generating function of the server's working time in a cycle. We also give the relationship between the discrete-time queueing system to its continuous-time counterpart. Finally,some examples and numerical results are presented.     

Discrete-time Geo/G/1 retrial queues with general retrial time and Bernoulli vacation

This paper considers a discrete-time Geo/G/1 retrial queue where the retrial time has a general distribution and the server is subject to Bernoulli vacation policy.It is assumed that the server, after each service completion,begins a process of search in order to find the following customer to be served with a certain probability,or begins a single vacation process with complementary probability. This paper analyzes the Markov chain underlying the queueing system and obtain its ergodicity condition.The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle,busy or on vacation.Finally,the author gives two stochastic decomposition laws,and as an application the author gives bounds for the proximity between the system size distributions of the model and the corresponding model without retrials.     

Many-server queues with customer abandonment: A survey of diffusion and fluid approximations

The performance of a call center is sensitive to customer abandonment.In this survey paper,we focus on G/GI/ n + GI parallel-server queues that serve as a building block to model call center operations.Such a queue has a general arrival process(the G ),independent and identically distributed(iid) service times with a general distribution(the first GI),and iid patience times with a general distribution(the +GI).Following the square-root safety staffing rule,this queue can be operated in the quality- and efficiency-driven(QED) regime,which is characterized by large customer volume,the waiting times being a fraction of the service times,only a small fraction of customers abandoning the system,and high server utilization.Operational efficiency is the central target in a system whose staffing costs dominate other expenses.If a moderate fraction of customer abandonment is allowed,such a system should be operated in an overloaded regime known as the efficiency-driven (ED) regime.We survey recent results on the many-server queues that are operated in the QED and ED regimes.These results include the performance insensitivity to patience time distributions and diffusion and fluid approximate models as practical tools for performance analysis.     

ANALYSIS AND COMPUTATIONAL ALGORITHM FOR QUEUES WITH STATE-DEPENDENT VACATIONS II： M（n）/G/1/K

We study a single-server queueing system with state-dependent arrivals and general service distribution, or simply M（n）/G/1/K, where the server follows an N policy and takes multiple vacations when the system is empty. We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. The only input requirements are the Laplace-Stieltjes transforms of the service time distribution and the vacation time distribution, and the state-dependent arrival rate. The computational complexity of the algorithm is O（K^3）.     

Optimal control of an M/G/1 retrial queue with vacations

In this note, we consider an M/G/1 retrial queue with server vacations, when retrial times, service times and vacation times are arbitrary distributed. The distribution of the number of customers in the system in stationary regime is obtained in terms of generating function. Next, we give heavy traffic approximation of such distribution. We show that the system size can be decomposed into two random variables, one of which corresponds to the system size of the ordinary M/G/1 FIFO queue without vacation. Such a stochastic decomposition property is useful for the computation of performance measures of interest. Finally, we solve simple problems of optimal control of vacation and retrial policies.     

Analysis of batch arrival queue with randomized vacation policy and an un-reliable server

This paper examines an M^[x]G/1 queueing system with an unreliable server and a delayed repair,in which the server operates a randomized vacation policy with multiple vacations.Whenever the system is empty,the server immediately takes a vacation.If there is at least one customer found waiting in the queue upon returning from a vacation,the server will be immediately activated for service.Otherwise,if no customers are waiting for service at the end of a vacation,the server either remains idle with probability p or leaves for another vacation with probability 1-p.Whenever one or more customers arrive when the server is idle,the server immediately starts providing service for the arrivals.The server may also meet an unpredictable breakdown and the repair may be delayed.For such a system the authors derive the distributions of some important system characteristics,such as the system size distribution at a random epoch and at a departure epoch,the system size distribution at the busy period initiation epoch,and the distribution of the idle period and the busy period.The authors perform a numerical analysis for changes in the system characteristics,along with changes in specific values of the system parameters.A cost effectiveness maximization model is constructed to explain the benefits of such a queueing system.     

Product Form Solution of a Queuing-Inventory System with Lost Sales and Server Vacation

In this study, the authors consider an M/M/1 queuing system with attached inventory under an(s, S) control policy. The server takes multiple vacations whenever the inventory is depleted.It is assumed that the lead time and the vacation time follow exponential distributions. The authors formulate the model as a quasi-birth-and-dearth(QBD) process and derive the stability condition of the system. Then, the stationary distribution in product form for the joint process of the queue length,the invent...     

ANALYSIS AND COMPUTATIONAL ALGORITHM FOR QUEUES WITH STATE-DEPENDENT VACATIONS Ⅱ:M(n)/G/1/K

We study a single-server queueing system with state-dependent arrivals and general service-distribution.or simply M(n)/G/1/K.where the server follows an N policy and takes multiple vacationswhen the system is empty.We provide a recursive algorithm using the supplementary variable tech-nique to munerically compute the stationary queue length distribution of the system.The only inputrequirements are the Laplace-Stieltjes transforms of the service time distribution and the vacation timedistribution.and the state-dependent arrival rate.The computational complexity of the algorithm isO(K~3).     

THE M/M/c QUEUE WITH （e,d） SETUP TIME

The authors present a new queueing model with （e, d） setup time. Using the quasi-birth-and-death process and matrix-geometric method, the authors obtain the stationary distribution of queue length and the LST of waiting time of a customer in the system. Furthermore, the conditional stochastic decomposition results of queue length and waiting time are given.     

The M/M/1 queue with working vacations and vacation interruptions

In this paper, we study the M/M/1 queue with working vacations and vacation interruptions. The working vacation is introduced recently, during which the server can still provide service on the original ongoing work at a lower rate. Meanwhile, we introduce a new policy：, the server can come back from the vacation to the normal working level once some indices of the system, such as the number of customers, achieve a certain value in the vacation period. The server may come back from the vacation without completing the vacation. Such policy is called vacation interruption. We connect the above mentioned two policies and assume that if there are customers in the system after a service completion during the vacation period, the server will come back to the normal working level. In terms of the quasi birth and death process and matrix-geometric solution method, we obtain the distributions and the stochastic decomposition structures for the number of customers and the waiting time and provide some indices of systems.     

Comparison of Gatekeeping and Non-gatekeeping Designs in a Service System with Delay-sensitive Customers

This paper is motivated by a concern in China's current medical practice in which patients can bypass the primary care and seek secondary care directly.We employ a queueing approach to examine two settings,i.e.the gatekeeping and non-gatekeeping settings,in a service system consisting of two types of service provider-one with basic skills(C-H),e.g.community hospital,and the other with advanced skills(AAA-H),e.g.specialist hospital.Customers are heterogeneous with respect to service requirement.The C-H can only serve customers with a complexity level of service requirement lower than the cure threshold,while the AAA-H can serve all customers.We aim to analyze the social planner's capacity decision for the C-H in both settings and assess the relative merit of each setting with respect to total social welfare,i.e.the sum of customer benefits net of customer delay costs and service providers7 operating costs.Our findings show that when the C-H's capacity is exogenous,the gatekeeper setting is preferable if the capacity of the C-H is in the intermediate range because customers'self-selection behavior gives rise to negative externality.When the C-H's capacity is optimized by the social planner,the non-gatekeeping setting is preferable if the capacity of the AAA-H is large or the cure threshold is high,because customers'self-selection behavior as well as the investment in the C-H's capacity can result in a better distribution of demand among the two service providers.The gatekeeping setting is preferable if the cure threshold is low because it is economical for the social planner to invest in a large capacity in the C-H to serve all customers.We also show numerically the conditions under which the two settings can achieve the first-best solution.     

Stationary Analysis and Optimal Control Under Multiple Working Vacation Policy in a GI/M~(a,b)/1 Queue

This paper considers an infinite buffer renewal input queue with multiple working vacation policy wherein customers are served by a single server according to general bulk service(a, b)-rule(1 ≤ a ≤ b). If the number of waiting customers in the system at a service completion epoch(during a normal busy period) is lower than ‘a', then the server starts a vacation. During a vacation if the number of waiting customers reaches the minimum threshold size ‘a', then the server starts serving this batch with a lower rate than that of the normal busy period. After completion of a batch service during working vacation, if the server finds less than ‘a' customers accumulated in the system, then the server takes another vacation, otherwise the server continues to serve the available batch with that lower rate. The maximum allowed size of a batch in service is ‘b'. The authors derive both queue-length and system-length distributions at pre-arrival epoch using both embedded Markov chain approach and the roots method. The arbitrary epoch probabilities are obtained using the classical argument based on renewal theory. Several performance measures like average queue and system-length, mean waiting-time, cost and profit optimization are studied and numerically computed.     

A Multi-indentured Repairable Inventory System with Limited and Different Diagnostic Repair Resources

This paper describes the characteristics of a single-base repairable inventory system and multi-indenture structure with communality and redundancy. At the base, there are a working field that has a maximum of z equipment on line and its repair channels, which mainly consist of two centers： line replaceable units （LRUs） diagnostic center and shop replaceable units （SRUs） repair center. In these two centers, the resources are finite and different. We introduce routing probabilities to express the repair relationships between resources and failed replaced units. The diagnostic time follows exponential distribution and SRU repair time follows general distribution with known average. We analyze the general model of this system, and present an approximate solution that uses two-step negative binomial approximation to obtain the expected backorders of all LRUs. The first step is fitting negative binomial distributions to the distributions of the number for all SitUs at the repair center, and the second is fitting negative binomial distributions to the convolutions of the distributions of the number for all LRUs in diagnostic and assemblydelayed. Thus, we give the approximate value of operational awilability of equipment based on this method. Finally, the results under two policies of routing probabilities are compared by the optimal curves.     

RANDOMIZED POLICY OF A POISSON INPUT QUEUE WITH J VACATIONS

<正>This paper studies the operating characteristics of an M/G/1 queuing system with a randomized control policy and at most J vacations.After all the customers are served in the queue exhaustively, the server immediately takes at most J vacations repeatedly until at least N customers are waiting for service in the queue upon returning from a vacation.If the number of arrivals does not reach N by the end of the J~(th) vacation,the server remains idle in the system until the number of arrivals in the queue reaches N.If the number of customers in the queue is exactly accumulated N since the server remains idle or returns from vacation,the server is activated for services with probability p and deactivated with probability(1-p).For such variant vacation model,other important system characteristics are derived,such as the expected number of customers,the expected length of the busy and idle period,and etc.Following the construction of the expected cost function per unit time,an efficient and fast procedure is developed for searching the joint optimum thresholds(N~*,J~*) that minimize the cost function.Some numerical examples are also presented.     

ANALYSIS OF A PRODUCTION-INVENTORY MODEL WITH ERLANGIAN DEMAND ARRIVAL PROCESS

We study a production-inventory system having a machine, a storage facility. The demand for the product is governed by an Erlangian demand arrival process, where demand sizes are independent and identically distributed random variables. A two-criticalnumber policy (m, M) is used to control a machine‘s setups and shutdowns, namely, a machine is shut down whenever the inventory level reaches M, and resumes operating only when the inventory level falls below the critical number m(m ≤ M). We obtain the steady state distribution of the inventory process and some performance measures of the process.     

Optimal policy for controlling two-server queueing systems with jockeying

This paper studies the optimal policy for joint control of admission, routing, service, and jockeying in a queueing system consisting of two exponential servers in parallel.Jobs arrive according to a Poisson process.Upon each arrival, an admission/routing decision is made, and the accepted job is routed to one of the two servers with each being associated with a queue.After each service completion, the servers have an option of serving a job from its own queue, serving a jockeying job from another queue, or staying idle.The system performance is inclusive of the revenues from accepted jobs, the costs of holding jobs in queues, the service costs and the job jockeying costs.To maximize the total expected discounted return, we formulate a Markov decision process(MDP) model for this system.The value iteration method is employed to characterize the optimal policy as a hedging point policy.Numerical studies verify the structure of the hedging point policy which is convenient for implementing control actions in practice.     

SERVICE LEVEL EVALUATION OF AN ASSEMBLY-TO-ORDER SYSTEM

An assembly-to-order system, which at the end the buffer distinguishes its assembly stages of the system from the downstream systems, is considered in this paper. The system produces semi-finished products for the downstream system and starts from a basic subassembly, and at each stage a component is assembled into the corresponding subassembly. The basic subassembly, components and buffer all follow a periodic-review, order up-to-level inventory policy. The buffer holds the semi-finished products to serve the specific demand from the downstream system. The service level of the system is determined by aggregate effects of the components held at stockpiles before the buffer and the basic subassembly. In order to measure the service level of the system, some notations and assumptions are made, on which the closed form expression of the service level of the system is achieved.     

Data-Driven Fitting of the G/G/1 Queue

The Maximum Likelihood Estimation(MLE)method is an established statistical method to estimate unknown parameters of a distribution.A disadvantage of the MLE method is that it requires an analytically tractable density,which is not available in many cases.This is the case,for example,with applications in service systems,since waiting models from queueing theory typically have no closed-form solution for the underlying density.This problem is addressed in this paper.MLE is used in combination with Stochastic Approximation(SA)to calibrate the arrival parameterθof a G/G/1 queue via waiting time data.Three different numerical examples illustrate the application of the proposed estimator.Data sets of an M/G/1 queue,G/M/1 queue and model mismatch are considered.In a model mismatch,a mismatch is present between the used data and the postulated queuing model.The results indicate that the estimator is versatile and can be applied in many different scenarios.