Abstract
A single-server queuing-inventory system in which arrivals are governed by a batch Markovian arrival process and successive arrival batch sizes form a finite first-order Markov chain is considered in this paper. Service is provided in batches according to a batch Markovian service process, with consecutive service batch sizes forming a finite first-order Markov chain. A service starts for the next batch on completion of the current service, provided that inventory is available at that epoch; otherwise, there will be a delay in starting the next service. When the service of a batch is completed, the inventory decreases by 1 unit, irrespective of batch size. A control policy in which the server goes on vacation when a service process is frozen until a quorum can initiate the next batch service is proposed to ensure idle-time utilization. During the vacation, the server produces inventory (items) for future services until it hits a specified level L or until the number of customers in the system reaches a maximum service batch size N, with whichever occurring first. In the former case, a server stays idle once the processed inventory level reaches L until the number of customers reaches (or even exceeds because of batch arrival) a maximum service batch size N. The time required for processing one unit of inventory follows a phase-type distribution. In this paper, the steady-state probability vector of this infinite system is computed. The distributions of inventory processing time in a vacation cycle, idle time in a vacation cycle, and vacation cycle length are found. The effect of correlation in successive inter-arrival times and service times on performance measures for such a queuing system is illustrated with a numerical example. An optimization problem is considered. The proposed system is then compared with a queuing-inventory system without the Markov-dependent assumption on successive arrivals as well as service batch sizes using numerical examples.
Highlights
The server goes on vacation if there are not enough customers in queue to initiate the batch service
In Model I, the server goes on vacation once there are not enough customers to initiate the batch service as specified by the Markov chain rule for service batch sizes
This increases the number of inventory processed as well as the idle time compared to Model II
Summary
The stage of development had a relaxed assumption of independence between successive inter-arrival times and/or successive service times. (Refer to [2,3,4] for more details and to [5,6,7] for reviews on BMAP.) Successive arrival batches are assumed to be mutually independent and “within independent” in the sense that successive arrival batch sizes are independent. This is true for successive service batches. Successive arrival batch sizes and successive service batch sizes are determined by two distinct Markov chains
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