Zero-Inventory Conditions For a Two-Part-Type Make-to-Stock(19)

We consider the dynamic scheduling of a two-part-type make-tostock production system using the model of Wein [12]. Exogenous demand for each part type is met from finished goods inventory; unmet demand is backordered. The control policy determines which pa

Appendix

AProbabilityofNoLowPriorityCustomerInthissection,wederivethestationaryprobabilityofhavingnolowprioritycustomersinapriorityM/G/1queuewithtwoclassesofcustomersandpreemptiveresume.Customersofclass1and2arriveaccordingtoPoissonprocesseswithparametersλ1andλ2,respectively.Class1hasthehighestpriority.Theservicetimeshavearbitrarydistributionswithmean1/µ1and1/µ2.Letγ2=P(X2=0),λ=λ1+λ2andρ=λ1/µ1+λ2/µ2.

FromCorollary1ofKeilsonandServi[7]thepgfofthenumberofclass2customersinthesystemis:

πS2(u)=

where

isthepgfofthenumberofclass2customersinthesystem

giventhatnoclass2customerisinservice

αT2istheLaplacetransformofT2,theservicetimeofclass2σBP1istheLaplacetransformoftheclass1busyperiod 2,thetimefromthebeginningofserviceofclass2istheLaplacetransformofTαT 2=αT2(s+λ1 λ1σBP1(s))untilthecustomerleavesthesystem,αT 2 2]=λ2E[Tρ 2 2]).(s)=(1 αT(λ2))/(sE[TαT 22πB2

Wewillusethefactthat

γ2(λ2)αT 2πB2(0).=πS2(0)=(1 ρ 2) 1 ρ 2αT(λ)2 2(30)(λ2 λ2u)(1 ρ 2)αT 2πB2(u), 1 ρ 2αT(λ λu)22 2(29)

ButfromKeilsonandServi[7],wealsohave,

πB2(0)=

ρ 21 ρ1[λ λ1σBP1(λ2)]λ2ρ2=.1 ρ1

19(31)(32)