A study on bulk arrival non Markovian single service queue with state dependent service

Abstract

A M^x/G/1 Queueing system with two service modes under (a, c, b) policy is considered. A single server Poisson queue with state dependent service is considered for analysis. The service is rendered with different service patterns (different distributions) as singly or in bulk according to the queue length. If the queue length reaches the threshold value ‘a’ then the server does single service and if the queue length reaches the value ‘c’(c > a), then the server selects batch service with variable batch size with a maximum batch size of ‘b’ (b > c). The server starts the service only if number of customers in the queue is at least ‘a’. The Server switches over from single service to batch service or vice-versa only at service initiation epochs depending on the queue length. The probability generating function of the queue size at an arbitrary time epoch is obtained using supplementary variables technique. Some important performance measures are also obtained. Numerical example is presented in order to illustrate the effectiveness of the model.

Appendix

1.1 Proof of the theorem 5.3.1

Let I _j = 1, if the system states becomes j during idle period= 0, otherwise.

$$ {\hbox{Let P}}\left( {{{\hbox{I}}_{\rm{j}}} = {1}} \right) = {\pi_{\rm{j}}},\,\,{\hbox{j}} = 0,{1},{2},{3} \ldots, {\hbox{a}} - {1} $$

(A-1)

Define $\pi_{\rm{j}}^1,\,\,\pi_{\rm{j}}^2$ as the probability that there are ‘j’ customers in the queue at batch service completion and single service completion epochs respectively.

Conditioning on the queue size at service completion epochs, for batch service completion epoch,

$$ \pi_{\rm{j}}^1 = {\alpha_{\rm{j}}} + \sum\limits_{{\rm{k}} = 0}^{\rm{j}} {\,{\alpha_{\rm{k}}}{\hbox{P}}\left( {{\hbox{I}}_{{\rm{j}} - {\rm{k}}}^\prime = {1}} \right),\quad \quad {\hbox{j}} = {1},{2},{3},...,{\hbox{a}} - {1}} $$

(A-2)

where ${\hbox{P}}\left[ {{\hbox{I}}_{\rm{j}}^\prime = {1}} \right]$ is the probability that the system state becomes ‘j’ when the server is in idle period and α _j = p_j is the probability that ‘j’ customers are in the queue at batch service completion epoch.

$$ {\hbox{Let P}}\left[ {{\hbox{I}}_{\rm{j}}^\prime = {1}} \right] = {\Phi_{\rm{j}}}\,\,{\hbox{and}}\,\,{\Phi_0} = {1} $$

$$ {\hbox{Where}}\,\quad \quad {\Phi_{\rm{j}}} = \sum\limits_{{\rm{k}} = 1}^{{\rm{j}} - 1} {{{\hbox{g}}_{\rm{k}}}{\Phi_{{\rm{j}} - {\rm{k}}}}\quad \quad {\hbox{j}} = {1},{2},{3}, \ldots } $$

(A-3)

For single service completion epoch, $\pi_{\rm{j}}^2 = {\gamma_{\rm{j}}}$, j = a−1, where γ _j is the probability that j (= a−1) customers are present in the queue at single service completion epoch.

$$ {\hbox{Thus}}\,\quad \quad {\pi_{\rm{j}}} = \pi_{\rm{j}}^1,\,{\hbox{j}} = 0,{1},{2}, \ldots, \,{\hbox{a}} - {2} $$

$$ {\pi_{{\rm{a}} - {1}}} = \pi_{{\rm{a}} - {1}}^1 + \pi_{{\rm{a}} - {1}}^2 $$

(A-4)

π _j (j = 0,1,2,…,a−1) can be computed using α _j (= p_j), γ _a−1 (= R_a−1) from the Eqs. 5–7.

From the above statement, $\sum\limits_{{\rm{j}} = 0}^{{\rm{a}} - 1} {}$ I_j, j the number of states visited during idle period,thus,

$$ {\hbox{E}}\left( {\sum\limits_{{\rm{j}} = 0}^{{\rm{a}} - 1} {{{\hbox{I}}_{\rm{j}}}} } \right) = \sum\limits_{{\rm{j}} = 0}^{{\rm{a}} - 1} {{\hbox{E}}\left( {{ }{{\hbox{I}}_{\rm{j}}}} \right) = \sum\limits_{{\rm{j}} = 0}^{{\rm{a}} - 1} {{\hbox{P}}\left( {{{\hbox{I}}_{\rm{j}}} = {1}} \right) = } } \sum\limits_{{\rm{j}} = 0}^{{\rm{a}} - 1} {{\pi_{\rm{j}}}} $$

(A-5)

Since the average staying time in any state during an idle period is 1/λ, the expected length of idle period E(I) is ${1}/\lambda \sum\limits_{{\rm{j}} = 0}^{{\rm{a}} - 1} {{\pi_{\rm{j}}}}$.

Hence the theorem.