Improved Priority Scheme for Unreliable Queueing System

In this paper, we consider a continuous review perishable inventory system with multi-server service facility. In such systems the demanded item is delivered to the customer only after performing some service, such as assembly of parts or installation, etc. Compared to most inventory models in which inventory is depleted at the demand rate, it is depleted, in this model, at the rate that the service occurs. We assume that the arrivals of customers are according to a Markovian arrival process (MAP) and that service time has exponential distribution. The ordering policy is based on (s; S) policy. The lead time is assumed to have exponential distribution. The customer who finds either all servers are busy or no item (excluding those in service) is in the stock, enters into an orbit of infinite size. These orbiting customers send requests at random time points for possible selection of their demands. The interval time between two successive request-time points is assumed to have exponential distribution. In addition to the regular customers, a second flow of negative customers following an independent MAP is also considered which will remove one of the customers in the orbit. The joint probability distribution of the number of busy servers, the inventory level and the number of customers in the orbit is obtained in the steady state. Various measures of stationary system performance are computed and the total expected cost per unit time is calculated.

This paper explores a stochastic queuing-inventory system designed to efficiently manage both new and returned items, offering practical solutions for diverse customer needs. The system differentiates services for various customer classes through dedicated channels, encompassing the sale of new and returned items, as well as the purchase of used items from customers. This model incorporates four parallel queues for distinct customer classes, each serviced by a dedicated server. Customer arrivals are modeled using a Markovian Arrival Process (MMAP), with service times being exponentially distributed and independent. An $(s, S)$ policy is implemented for replenishing fresh items. The primary objective is to enhance service accessibility for various customer types within a single facility, promoting operational efficiency and customer satisfaction. Additionally, the system's ability to purchase used items underscores its role in fostering sustainability in an evolving society. By applying the Neuts matrix geometric technique, the paper analyzes the system to derive the long-term probability distribution and significant performance metrics. Numerical methods are utilized to investigate key system parameters and performance measures, and a cost function is introduced and optimized concerning the reorder level. This comprehensive analysis offers valuable insights for optimizing inventory and queuing systems in practical, real-world applications.

A single-server queueing system with a marked Markovian arrival process of heterogeneous customers is considered. Type-1 customers have limited preemptive priority over type-2 customers. There is an infinite buffer for type-2 customers and no buffer for type-1 customers. There is also a finite buffer (stock) for consumable additional items (semi-products, half-stocks, etc.) which arrive according to the Markovian arrival process. Service of a customer requires a fixed number of consumable additional items depending on the type of the customer. The service time has a phase-type distribution depending on the type of the customer. Customers in the buffer are impatient and may leave the system without service after an exponentially distributed amount of waiting time. Aiming to minimize the loss probability of type-1 customers and maximize throughput of the system, a threshold strategy of admission to service of type-2 customers is offered. Service of type-2 customer can start only if the server is idle and the number of consumable additional items in the stock exceeds the fixed threshold. Stationary distributions of the system states and the waiting time are computed. In the numerical example, we show some interesting effects and illustrate a possibility of application of the presented results for solution of optimization problems.

A multi-server perishable inventory system with negative customer

We analyze a discrete-time queueing model where two types of customers, each having their own dedicated server, are accommodated in one single FCFS queue. Service times are deterministically equal to s≥1 time slots each. New customers enter the system according to a general independent arrival process, but the types of consecutive customers may be nonindependent. As a result, arriving customers may (or may not) have the tendency to cluster according to their types, which may lead to more (or less) blocking of one type by the opposite type. The paper reveals the impact of this blocking phenomenon on the achievable throughput, the (average) system content, the (average) customer delay and the (average) unfinished work. The paper extends the results of earlier work where either the service times were assumed to be constant and equal to 1 slot each, or the customers all belonged to the same class. Our results show that, in case of Poisson arrivals, for given traffic intensity, the system-content distribution is insensitive to the length (s) of the service times, but the (mean) delay and the (mean) unfinished work in the system are not. In case of bursty arrivals, we find that all the performance measures are affected by the length (s) of the service times, for given traffic intensity.

We consider Markovian multi-server queues with two types of impatient customers: high- and low-priority ones. The first type of customer has a non-preemptive priority over the other type. After entering the queue, a customer will wait a random length of time for service to begin. If service has not begun by this time he or she will abandon and be lost. We consider two cases where the discipline of service within each customer type is first-come first-served (FCFS) or last-come first-served (LCFS). For each type of customer, we focus on various performance measures related to queueing delays: unconditional waiting times, and conditional waiting times given service and given abandonment. The analysis we develop holds also for a priority queue with mixed policies, that is, FCFS for the first type and LCFS for the second one, and vice versa. We explicitly derive the Laplace–Stieltjes transforms of the defined random variables. In addition we show how to extend the analysis to more than two customer types. Finally we compare FCFS and LCFS and gain insights through numerical experiments.

This paper analyses a production inventory model with varying production rates where demand is according to Marko­vian arrival process (MAP). The time between additions of two successive items by production to the inventory follows Phase type distribution. When the inventory level decreases to a predetermined level production starts and the rate of production is higher until the number of items in the inventory crosses that level. The arriving customers are directed to a buffer of finite capacity equal to the current inventory level and service times are exponentially distributed. An arriving customer who notices the buffer full, proceeds to an orbit of infinite capacity and retry from there. Inter-retrial times follow an exponential distribution. Various system performance measures of the model are defined. A suitable cost function is also constructed.

In this paper we simulate a queueing model useful in a service system with the help of ARENA simulation software. The service calls (henceforth referred to as customers) arrive to a processing center according to a Markovian arrival process (MAP). There is a buffer of finite size to hold the customers. Any customer finding the buffer is considered lost. An arriving customer belongs to one of three types, and the admitted customer is served by one of many dedicated servers (exclusively set aside for each of the three types of customers) or by one of many flexible servers who are capable of servicing all types of customers. The flexible servers are used only when the respective dedicated servers are all busy. A priority scheme is used to select the type of customer from the buffer when a flexible server is called for servicing the waiting customers. The processing times are assumed to be of phase type. Simulated results are discussed.

A multiserver queueing system with infinite and finite buffers, two types of customers, and two types of servers as a model of a call center with a call-back for lost customers is investigated. Type 1 customers arrive to the system according to a Markovian arrival process. All rejected type 1 customers become type 2 customers. Typer,r=1,2, servers serve typercustomers if there are any in the system and serve typer′,r′=1,2, r′≠r,customers if there are no typercustomers in the system. The service times of different types of customers have an exponential distribution with different parameters. The steady-state distribution of the system is analyzed. Some key performance measures are calculated. The Laplace-Stieltjes transform of the sojourn time distribution of type 2 customers is derived. The problem of optimal choice of the number of each type servers is solved numerically.

We analyze and compare three (s,S) inventory systems with positive service time and retrial of customers. In all of these systems, arrivals of customers form a Poisson process and service times are exponentially distributed. When the inventory level depletes to s due to services, an order of replenishment is placed. The lead-time follows an exponential distribution. In model I, an arriving customer, finding the inventory dry or server busy, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the inventory dry or server busy, returns to the orbit with probability δ and is lost forever with probability (1−δ). In addition to the description in model I, we provide a buffer of varying (finite) capacity equal to the current inventory level for model II and another having capacity equal to the maximum inventory level S for model III. In models II and III, an arriving customer, finding the buffer full, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the buffer full, returns to the orbit with probability δ and is lost forever with probability (1−δ). In all these models, the interretrial times are exponentially distributed with linear rate. Using matrix-analytic method, we study these inventory models. Some measures of the system performance in the steady state are derived. A suitable cost function is defined for all three cases and analyzed using graphical illustrations.

We consider a single server retrial queueing model with two types of customers, abandonment of customers and search for priority customers. Two types of customers namely type I and type II arrive according to two independent Poisson processes and upon arrival, if the server is busy, join orbits I and II respectively. Orbit I is of finite capacity while orbit II is of Infinite capacity. Retrial attempts are made by both types of customers at constant rates and each unsuccessful retrial attempt is followed by abandonment of customers from the respective orbits with some probability. Type I customer has higher priority over type II customers. Search for customers in orbit 1 is introduced to reduce their waiting time and also to minimize the loss of such customers due to abandonment. Condition for stability is established and the system state distribution is computed. Several performance measures of the system that influences the efficiency are derived and are illustrated graphically/numerically.

This paper considers a discrete time priority queue formed by the customers induced interruption during the service time. The arrivals of customers are modeled by the Markovian Arrival Process (MAP) and the service time follows discrete phase type distribution. During the service, the customer may interrupt with a pre-assigned probability and the interrupted customers are moved to lower priority queue. The interrupted customers are not allowed any further interruptions except in the case of preemptive priority. Preemptive and non- preemptive disciplines to lower priority customers are considered. Matrix analytic method is used to analyse the model. The marginal distributions of both queue lengths in each discipline are analysed.

Obtaining exact blocking probabilities for tandem queues with finite capacities is not a trivial problem. In this paper, we propose a computational approximation method using max-plus algebra for computing blocking probability in a Poisson-driven 2-node tandem queue with finite capacities and constant service times. The blocking probability of a finite-capacity queueing system can be obtained from either the tail probability of stationary waiting time or the difference between two expected stationary waiting times at the first node of the corresponding extended 3-node tandem queue. The computational results in this study show that the proposed approach provides a good approximation of the blocking probability, and in particular, it works well under moderately to heavily loaded situations. The proposed approach is not limited to a particular blocking policy, system structure, or service time; hence, it is applicable to general queues with finite buffer capacities and various blocking policies.

This paper considers the comparison of two left( {s,S} right) production inventory systems with retrials of unsatisfied customers. The time for producing and adding each item to the inventory is exponentially distributed with rate beta. However, a production rate alpha beta higher than beta is used at the beginning of the production. The higher production rate will reduce customers’ loss when inventory level approaches zero. The demand from customers is according to a Poisson process. Service times are exponentially distributed. Upon arrival, the customers enter into a buffer of finite capacity. An arriving customer, who finds the buffer full, moves to an orbit. They can retry from there and inter-retrial times are exponentially distributed. The two models differ in the capacity of the buffer. The aim is to find the minimum value of total cost by varying different parameters and compare the efficiency of the models. The optimum value of alpha corresponding to minimum total cost is an important evaluation. Matrix analytic method is used to find an algorithmic solution to the problem. We also provide several numerical or graphical illustrations.

This paper studies a queueing system with a Markov arrival process with marked arrivals and PH-distribution service times for each type of customer. Customers (regardless of their types) are served on a mixed first-come-first-served (FCFS) and last-come-first-served (LCFS) nonpreemptive basis. That is, when the queue length is N (a positive integer) or less, customers are served on an FCFS basis; otherwise, customers are served on an LCFS basis. The focus is on the stationary distribution of queue strings, busy periods, and waiting times of individual types of customers. A computational approach is developed for computing the stationary distribution of queue strings, the mean of busy period, and the means and variances of waiting times. The relationship between these performance measures and the threshold number N is analyzed in depth numerically. It is found that the variance of the virtual (actual) waiting time of an arbitrary customer can be reduced by increasing N . c © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 399–421, 2000