Unreliable bulk queue with single vacation and an early startup time

Extract

[...] of system size of the H V + H I + H U + H D unreliable Mx/M/1 queueing system by J.C.Ke and ψ = 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 α 2 α λ + + + 2 + 2 β β µ(1 ρ1 ) Proof : N 1 m m nψ n + nφn + γ ψ n + λE(x) γ 2 + γη + η2 λ n n n m N ψ n + φn + + n γ η n From Equation we obtain d P x dz unreliableM / M NPolicy ( ) z + d (ψ z dz where d P x dz unreliableM / M NPolicy ( ) z = E(x(x α 2 α λ + + + 2 β 2 β µ(1 ρ1 ) N 1 m m nψ n + nφn + γ ψ n + λE(x) γ 2 + γη + η2 λ n n n m N ψ n + φn + + λ n n γ η 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 [λ(1 + η Similarly if we multiply by z and by zn ) and adding over all possible values of n we obtain H Br = αH W λ(1 + β We apply the similar procedure to equations - to obtain the following equation (λ + γ λx(z))H D = ηH V + λ 1)H I By defining π n = g i πn i n with π0 = 1 and ψ n = α i πn i n and following the arguments as in Lee et al we get from equations & that H I = µPW m ψ n zn λ n subsequently equation becomes H D = µPW m η ψ n z n where = λ(1 λ n ( γ + η + Equation can be rewritten as µP H D = W h k z k h z r r γ 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 η ψn zn λ n η + Solving equations and recursively we obtain λPU = γ PD (k)πn k k n where PD is the coefficient of zk of HD(z) (i.e.) PD(k) = µPW k h i h k γ i=m n n 1 Hence λPU = µPW h r h k πn k r k This can be written as where φ(1) = n Thus n n λPU = µPW (1)φn π r=m r k k N n n λH U = µPW φ(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 γ η ψn zn λ φn zn λ n γ + η + α µ(z + zw(z) + β + If denotes the total p.g.f. [...]

[...] J.C.Ke has considered a batch arrival single server queueing system in which an unreliable server operates policy and takes vacation. He has considered multiple vacation policy for his discussion. In his paper he has applied a property of stochastic decomposition to develop the distribution of number of customers in the system and derived the system characteristics. In the present paper we analyze the bi level control policy for a batch arrival single unreliable server queueing system in which the sever takes a single vacation when the system becomes empty. [...]

[...] By considering γ = µ = λ = 0.4 and varying η η L Tc(N) PV Table 2 By letting η = µ = λ = 0.4 and varying γ γ L Tc(N) PI PU PD Table 3 By substituting η = γ = λ = 0.4 and varying µ µ L Tc(N) PW Table 4 By considering η = γ = µ = 1.2 and varying λ λ L Tc(N) ρ From table 1,2 and it is found that the expected system size and the total cost function decrease as the vacation rate service rate and the startup parameter(γ) increase and conversely table indicates that L and TC(N) increase as the arrival rate increases. Based on this analysis an optimal size N can be determined for this model. Conclusion : A bulk arrival , unreliable server queue with early startup and single vacation has been analysed using p.g.f. This method works efficiently for deriving the steady state probabilities of the model considered. [...]

[...] The customers who arrive and find that the server is busy ( or) in breakdown must wait in the queue until the server is available. Once the broken-down server is repaired, he immediately returns to serve the customers until the system becomes empty. Although no service occurs during the repair period of a broken-down state, customers continue to arrive according to the compound Poisson process. The vacation time startup time service time and repair time ( R ) are assumed to be independent of each other and follow exponential distributions with parameters η, γ, µ and β respectively. [...]