Optimization Of Queueing Model

-in thepaper, we are considering the single server queueing system have interdependent arrival of the service processes having bulk service.In this article, we consider that the customers are served k at any instance except when less then k are in the system &ready to provide service at which time customers are served. Keyword: -Interdependent queueing models, arrival process, service process, waiting line system, mean dependence. 1. OPTIMIZATION M/MK/1 QUEUEING MODEL WITH VARYING BATCH SIZE :In this type of systems, the interdependence could be induced by considering the dependent structure with parameters λ , μ and ∈as marginal arrival rate , service rate and mean dependence rate respectively . Let Pn(t) be the probability when there are n customers in system at time t . The difference – differential equations of above modelmay have written as, P′n(t) = −( λ + μ − 2 ∈)Pn(t) + (λ−∈)Pn−1(t) + Pn−k(t); n ≥ 1 P′0(t) = −(λ−∈)P0(t) + (μ−∈) ∑ Pi(t) k i=1 .................. (1) Let us considerthat, the system achieved the steady state, therefore the transition equations of considered model ar, −( λ + μ − 2 ∈)Pn + (λ−∈)Pn−1 + (μ−∈)Pn−k = 0 ; n ≥ 1 −(λ−∈)P0 + (μ−∈) ∑ Pi k


OPTIMIZATION / [ ] / QUEUEING MODEL WITH VARYING BATCH SIZE :-
In this type of systems, the interdependence could be induced by considering the dependent structure with parameters , and ∈as marginal arrival rate , service rate and mean dependence rate respectively .

MEASURES OF EFFECTIVENESS: -
The probability that the system is empty is, ……………………. (5) Where is as given in equation (3). From tables (5.1), (5.2) and equation number---(5), we observe that for fixed of , and ∈ , the value of 0 increases with respect to increase in . As the dependence parameter∈increases the value of 0 increases for fixed values of , and .The value of 0 decreases for fixed values of , and ∈ as increases. As increases the valueof 0 increases for fixed values of the , and dependence parameter∈ . If the mean dependence rate, is zero then the value of 0 is also same as in the / [ ] /1 − . The average no. of customers in the system can obtained as  (7) where is as given in equation (3). The value of and has been computed and given in tables-5.3 and table-5.5 for provided values of , and for different values ∈and respectively. The values of and for fixed values of ∈and & for varying and also given in tables-5.4 and table-5.6.
By equations 6 and 7, also for the corresponding tables we observe,that as∈increase, the values of and are decreasing and also as increases the values of and are decreasing for fixed values of other parameters. As the arrival rate increases, the values of and are increasing for fixed values of , and ∈. As increases the values of and are decreasing for fixed values of , and ∈. When the dependence parameter∈= 0 then the average queue length is same as that of / [ ] /1model. When = 1 this is same as / /1 interdependence model.
The variability of this model can be obtained as ………………. (8) where is as given in equation (3). The coefficient of variation of the model is  (6) and (7). The values of 'variability of system' and 'coefficient of variation'for various values of , ∈ forfixed values of , are computed which are given in tables-5.7& 5.9 . The values of 'variability of the system' and 'coefficient of variation' for fixed values of , ∈ and for various values of , are provided in tables (5.8) and (5.10).
From equation-9 a& from the corresponding table we can observe that as increases the "variability of the system size" decreases and "coefficient of variation" increases. As increases and for fixed values of , ∈ and , the 'variability of the system size' increases & the 'coefficient of variation decreases. We may observe that as ∈ increases the 'variability of the system size, decreases and 'coefficient of variation' increases for fixed values of , and . As increases, the 'variability of the system' decreases and the 'coefficient of variation increases'.
For this model ∈= 0 and = 1 reduces to / /1 classical model. The mean-queue length & 'variability of the system size' of this model are less than that of the classical. When = 1, this model becomes / / 1independent model for ∈= 0, this model is same as / [ ] /1model.