Journal of Probability and Statistics

Volume 2015, Article ID 505082, 8 pages

http://dx.doi.org/10.1155/2015/505082

## An Production Inventory Controlled Self-Service Queuing System

Department of Mathematics, National Institute of Technology Calicut, Kerala, Calicut 673601, India

Received 10 June 2015; Revised 17 September 2015; Accepted 27 September 2015

Academic Editor: Ramón M. Rodríguez-Dagnino

Copyright © 2015 Anoop N. Nair and M. J. Jacob. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider a multiserver Markovian queuing system where each server provides service only to one customer. Arrival of customers is according to a Poisson process and whenever a customer leaves the system after getting service, that server is also removed from the system. Here the servers are considered as a standard production inventory. Behavior of this system is studied using a three-dimensional QBD process. The condition for checking ergodicity and the steady state solutions are obtained using matrix analytic method. Unlike classical queuing models, the number of servers varies in this model according to an inventory policy.

#### 1. Introduction

In recent years, manufacturing commodities or inventories in anticipation of demand has fallen out of favor considerably. Most of the manufacturers produce items in response to the actual demand. The amount of time taken for the production of an item and the time at which a production order is to be placed are two important parameters in this regard. The standard production inventory is an answer to these parameters which is quite different from the usual inventory policies. Most of the studies consider inventory systems in which as many items as needed can be replenished all at once. But in production inventory, the replenishment is done on an item-by-item basis. That is, at the time at which the inventory falls to , the production process is switched on and is switched off at the time instance at which the on-hand inventory reaches . The production time of an item in the inventory is a random variable with a probability distribution. In production inventory, to know the status of the system both the inventory level and the production status (whether production process is on or off) must be known.

In classical inventory management problems, the service time of an inventory was assumed to be negligible. Inventory models with positive service time have first been investigated by Sigman and Simchi-Levi [1]. Thereafter a lot of works have been carried out in this area. For further details on inventory models with positive service time we refer to a survey article by Krishnamoorthy et al. [2]. It can be observed that most of the models discussed in the literature have considered queues in which a single server services an inventory. Inventories with/without lead time, perishable inventory, and so forth and attached with a server are also examined with different ordering policies such as and other general randomized policies. The studies of Parthasarathy and Vijayalakshmi [3], Deepak et al. [4], Manuel et al. [5], and Narayanan et al. [6] are some of the examples of the above said models.

In the case of production inventory with positive service time the work of Krishnamoorthy and Narayanan [7] is the first reported work. They have considered queue attached with a production inventory. The time for producing an item in the inventory follows a Markovian production scheme.

As mentioned above, most of the works in the literature consider systems where a single server is providing service to arriving customers. Nowadays, self-service inventory facilities are on the rise to get the customer satisfaction. It can be seen that there are many situations where the customer himself services inventory as in retail supermarket or online shopping and so forth. But studies on such self-service inventories are very less. In a self-service facility, there is no server to serve the items but one or more customers get service simultaneously depending on the availability of the inventory. That is, customers themselves serve the inventory and the time for serving the inventory can be considered as service time. In particular, we can think about a self-serviceable retail outlet which sells a particular type of inventory or an online shopping site exclusively for a single type of inventory. Such a system has the characteristics of a multiserver queuing system.

A self-service queuing system with positive service time was first investigated by Anoop et al. [8]. They have considered a standard inventory without lead time as servers and obtained the steady state probabilities, conditional distributions on the system size and inventory level, distribution of the inventory cycle time, and an optimization problem which optimizes the reorder quantity. Nair and Jacob [9] have analyzed a queuing system with retrial of demands by considering the inventory as servers.

Production inventory is very relevant when the manufacturer himself is meeting the demands. It could be a street vendor preparing snacks based on the demand for the item. Also it can be the manufacturer of customized items which will be produced depending on the actual demand. In the present work, we try to address a self-service inventory where the inventory is the production inventory. We consider the production inventory as servers of the queuing system. Unlike in classical queuing models, the number of servers varies in this model according to an inventory policy. The aim of this paper is to study simplest Markovian queuing system with varying number of servers which replenished according to an production inventory. It can be observed that there is a strong dependence between the system size and the inventory level. Therefore, a product form solution is not anticipated as described in Krishnamoorthy and Viswanath [10]. We use matrix analytic methods to find the steady state probabilities.

#### 2. Model Description

We consider a queuing system where arrival of customers (demand for an item) follows a Poisson process with rate . We have a standard production inventory as servers and the service time follows an exponential distribution with parameter . Therefore, when there are customers and items in the inventory, the effective service rate of the system is . After service completion an item in the inventory as well as a customer leaves from the system which results in the decrement of the system size and the number of servers (inventory). When the inventory level reaches , the production process is switched on and it is switched off when the inventory level reaches . The time required to produce an item in the inventory is assumed to be exponentially distributed with parameter . An arriving customer joins the system with probability 1 if he finds a free server (inventory). But whenever the number of customers exceeds or is equal to the available inventory, an arriving customer joins the system with probability and he leaves the system without waiting with probability .

We analyze the system by considering a three-dimensional QBD process , where represents the number of customers in the system at time , represents the inventory level at time , and represents status of the production process. The state space of the above continuous time Markov chain is given by for . Further indicates that the production process is off and indicates that the production process is on.

Proposition 1. *The infinitesimal generator of the QBD process is given by where the dimension of each block matrix is and the block matrices give the transition rates which are described as follows: *(a)*, , represents the transition rate matrices due to service completion at level (see Table 1).*(b)*, , represent the transition rate matrices which leave the first coordinate fixed at level (see Table 2).*(c)*, , represent the transition rate matrices due to arrival of customers at level (see Table 3).*