# Unreliable bulk queue with single vacation and an early startup time

- Abstract
- Introduction
- Model description
- Steady state results
- Probability generating function

- Characteristics of the model
- Optimal management policy
- Numerical examples
- Conclusion
- References

This paper studies the batch arrival queuing system under bi level control policy where an unreliable server operates an (m, N) policy with early startup and takes a single vacation whenever the system becomes empty. The probability generating function of the steady state system size probabilities is obtained in a closed form and the system characteristics are derived for the model. The stochastic decomposition property is also discussed. A cost model is developed to determine the optimal operating (m, N) policy in order to minimize the total expected cost. The optimal control of queuing system has achieved considerable attention in literature. Lee and Park [2] were first to examine the M/G/1 queuing system and later MX/G/1 queuing system under the bi level control with servers vacation .In their models, it is assumed that the server start his startup operation, when the queue length reaches or exceeds m (m

[...] of system size of the H V + H I + H U + H D unreliable Mx/M/1 queueing system by J.C.Ke and ? = H V + H I + H U + H D gives the conditional p.g.f of queue length during idle period (vacation period, buildup period, dormant period, breakdown period). Hence the well known stochastic decomposition property also holds good. Characteristics of the Model : In this section we derive some other performance measure. Let L denote the expected number of customers in the system for this model then L can be expressed as L = E(x(x ? 2 ? ? + + + 2 + 2 ? ? µ(1 ?1 ) Proof : N 1 m m n? n + n?n + ? ? n + ?E(x) ? 2 + ?? + ?2 ? n n n m N ? n + ?n + + n ? ? n From Equation we obtain d P x dz unreliableM / M NPolicy ( ) z + d (? z dz where d P x dz unreliableM / M NPolicy ( ) z = E(x(x ? 2 ? ? + + + 2 ? 2 ? µ(1 ?1 ) N 1 m m n? n + n?n + ? ? n + ?E(x) ? 2 + ?? + ?2 ? n n n m N ? n + ?n + + ? n n ? ? Hence we get L as in equation (25). [...]

[...] of PV(n) and PBr(n) respectively for z as follows: H I = PI n m n ; H U = PU n N n ; H D = PU n n N H W = PW n n ; H V = PV n n ; H Br = PBr n n and = z n n n By multiplying equations & by suitable powers of z and adding we obtain H V = µPW [?(1 + ? Similarly if we multiply by z and by zn ) and adding over all possible values of n we obtain H Br = ?H W ?(1 + ? We apply the similar procedure to equations - to obtain the following equation (? + ? ?x(z))H D = ?H V + ? 1)H I By defining ? n = g i ?n i n with ?0 = 1 and ? n = ? i ?n i n and following the arguments as in Lee et al we get from equations & that H I = µPW m ? n zn ? n subsequently equation becomes H D = µPW m ? ? n z n where = ?(1 ? n ( ? + ? + Equation can be rewritten as µP H D = W h k z k h z r r ? k r where hk represents the probability that k customers arrive during the startup time and hr be the coefficient of of z of r m ? ?n zn ? n ? + Solving equations and recursively we obtain ?PU = ? PD (k)?n k k n where PD is the coefficient of zk of HD(z) (i.e.) PD(k) = µPW k h i h k ? i=m n n 1 Hence ?PU = µPW h r h k ?n k r k This can be written as where ?(1) = n Thus n n ?PU = µPW (1)?n ? r=m r k k N n n ?H U = µPW ?(1) z n n Next to calculate HW(z) we multiply the equations & by corresponding powers of z and adding over n=1to we get, H W = N zµPW ? ? ?n zn ? ?n zn ? n ? + ? + ? µ(z + zw(z) + ? + If denotes the total p.g.f. [...]

[...] This is a part of the research work carried out to analyse breakdown queueing models with early startup and a single vacation. To know more about the system behavior, performance of measures and the sensitivity analysis are also carried out. References: 1. Lee .H.S. and M.M. Srinivasan (1989) ?Control policies for [...]