Abstract
This paper deals with the development and analysis of K-node series and parallel queueing model with load dependent service rates. Here it is assumed that the customers arrive to the initial queue and waiting line for service. After completing the service at first service station they may join one of the (K-1) queues which are parallel and connected to first queue in series. After getting service from the service station they leave the system. Here it is assumed that the service rates in each service station are dependent on number of customers in the queue connected to it .The arrival and service completions in each queue are assumed to follow Poisson processes. Using difference-differential equations the joint probability function of number of customers in each queue are derived. The system performance measures such as average number of customers in each queue ,throughput of each service station ,the probability of idleness of each server ,the waiting time of customer in each queue are derived explicitly. The sensitivity of the model with respect to parameters are analysed through numerical illustration. It is observed that the state dependent service rates has significant influence on performance measures. This model also includes the earlier models as particular cases for specific values of the parameters. This model is useful in analysing the communication networks , transportation systems ,production processes and cargo handling.
Highlights
A Queue is a waiting line of units demanding service at a server facility
Queueing models formulate a prerequisite for design and and development of several systems arising at places like Communication networks, ATM scheduling,Transportation systems, Production processes etc.,(Boxima O.J., [ ], Bunday.B.D., [ ], SrinivasaRaoet.al., [ ] [ ] [ ], Charan Jeet Singh et al, [ ],Ramasundari et al[ ])
In Tandem Queueing models the out put of one Queue formulate the input of the other.Jackson Paul[ ], Srinivasa Rao et al, [ ] [ ] [ ],Che Soong Kim et al[ ], Raghavendran et al[ ]and others have developed various tandem Queueing models with the assumption that arrivals and services are independent
Summary
A Queue is a waiting line of units demanding service at a server facility. Erlang [ ] pioneered the mathematical modelling of queueing system. Srinivasa Rao et al, [ ] [ ], Varma et al, [ ], Padmavathi et al, [ ], Nageswara Rao et al, [ ] Trinadha Rao et al, [ ], Suhasini et al, [ ] [ ] [ ], Rajasekhar Reddy et al [ ]have developed Queueing models with the assumption that the service rates are dependent on the number of customers in the Queue In all these papers they assumed that the nodes are connected in tandem and single. In several practical situations after getting service from the first Queue the customer may join one of the several Queues connected to it for service .For example in Communication networks after getting service from the first transmitter the data/voice packets are to be routed to one of the several buffers connected in parallel for forward transmission This type of scenario is visible in Production processes such as Glass manufacturing, where the raw material is converted as liquid glass. ).Multiplying equations (1)..(8)with probability generating function and summing over from 0 to we get Joint Probability generating function of number of customers in first, second,....kth buffers respectively at any time t as
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