Question 2.41. Please help! Reference: Fundamentals of Queueing Theory. Fourth Edition. Donald Gross; John F. Shortie; James M. Thompson; Freddie Mac Corporation; Carl M. Harris CHAPTER 2. SIMPLE MARKOVIAN QUEUEING MODELS https://irh.inf.unideb.hu/~jsztrik/education/16/Queueing_Problems_Solutions_2021_Sztrik.pdf

2.41. 112 An application of an iii/M/oo model to the eld of inventory control is as follows. A manufacturer ofa very expensive. rather infrequently demanded item uses the following inventory control procedure. She keeps a safety stock of .3 units on hand. The customer demand for units can be described by a Poisson process with mean A. Every time a request for a unit is made {a customer demand), an order is placed at the factory to manufacture another {this is called a oneforone ordering policy). The amount of time required to manufacture a unit is exponential with mean Up. There is a carrying cost for inventory on shelf of lift. per unit per unit time held on shelf (representing capital tied up in inventory which could be invested and earning interest. insurance costs, spoilage, etc.) and a shortage cost of $1) per unit [a shortage occurs when a customer requests a unit and there is none on shelf, i.e., safety stock is depleted to zero). It is assumed that customers who request an item but find that there is none immediately available will wait until stock is replenished by orders due in (this is called backordering or backlogging); thus one can look at the charge $3) as a discount given to the customer because he must wait for his request to be satised. The problem, then, becomes one of finding the optimal value of S that minimizes total expected costs per unit time; that is, nd the S that minimizes ElC] : hizpzl + pA 2 p(2) ($/unittime), ~=l z'Loo where z: is the steady-state on-hand inventory level (+ means items on shelf, means items in backorder} and 10(3) is the probability frequency function. Note that 2321 zplz) is the average value of the safety stock and /\\ 2:1230 pfz) is the expected number of backorders per unit time, since the second summation is the fraction oftime there is no on-shelf safety stock and /\\ is the average request rate. If p[z) could be determined. one could optimize EIC] with respect to S. {a} Show the relationship between Z and N, where N denotes the number of orders outstanding, that is, the number of orders currently being processed at the factory. Hence relate pfz) to Pn- SIMPLE MARKOVIAN QUEUEING MODELS (b) Show that the {pn} are the steady-state probabilities of an Art/,M/oo queue if one considers the orderprocessing procedure as the queueing system, State explicitly what the input and service mechanisms are. (c) Find the optimum S for A = Simonth, 1/,u. = 3 days, h L $503unit per month held. and p = $500 per unit backordered