Analysis of an M/M/1/∞ model with Bernoulli schedule during busy period, additional task, customers fluctuating priorities behavior during working vacation
Author(s): Rachna Rathore and RK Shrivastava
Abstract: This paper analyzes the customers (Clients) fluctuating priorities behavior during working vacation (WV) with Bernoulli scheduled busy period in an M/M/1/∞ model along with the assumption that some additional task apart from regular job is assigned to the server during working vacation. Customers arrive at the system with Poisson’s distributed rate λ and get served with exponentially distributed rate μ. During busy period (State) clients might enter the queue with probability r and could leave with probability q. After completing busy period, the server moves to a single working vacation with exponentially distributed rate γ where service is provided at slower pace than busy state as some additional task is assigned to server apart from regular job. In this paper we have introduced a new behavior of customers named as “Customers Fluctuating Behavior” during working vacation. According to this behavior customers standing in a queue wait for their turn might leave the queue with exponentially distributed rate ξ during working vacation not due to impatience or long queue or they have any issue with the system but it is due to their (Clients) fluctuating or can say confusing behavior on their priorities of doing work. If some clients are present at an epoch of WV completion, then the server return back to busy state with probability p otherwise do some secondary tasks with rate ϕ and probability m. During performing some secondary task server will not serve any client. If client arrives during this duration, then server will resume busy state with rate ψ. In this paper we have formulated various system probabilities in steady state condition by using PGF method. Various system measures such as average queue length in busy period, working vacation and in the system respectively, Mean sojourn time in the system, abandonment rate of clients in WV period is obtained explicitly. The impact of some parameters on some system measures has been illustrated numerically and graphically.
Rachna Rathore, RK Shrivastava. Analysis of an M/M/1/∞ model with Bernoulli schedule during busy period, additional task, customers fluctuating priorities behavior during working vacation. Int J Stat Appl Math 2024;9(5):129-136. DOI: 10.22271/maths.2024.v9.i5b.1825
