Question

Problem 2 There are three machines and two mechanics in a factory. The break time of each machine is exponentially distributed with A 1 (per day). The repair time of a broken machine is also exponentially distributed with a mean of 3 hours. (Mechanics work separately) (1). Construct the rate diagram for this queueing system. (be careful about the arrival rate A) (2). Set up the rate balance equations, then solve for pn's. (3). Compute L (4). Compute the actual mean arrival rate λ. (5). What does W, mean in this scenario? Compute it (in minutes).

Answer 1

Number of machines in the system: Number of servers or mechanics: Breaktime of each machine per day or Average arrival rate per day: C:2 X1 9 μ:--3 3 Average broken machine repared or average service rate per day(9hrs) per mechanic: Machine repair or service capacity per day -c41-6 Transition rate diagram: Possible states of the svstem: k:-0,1..6 6 Let TR steady-state probability of being in state k

π=[π π π π π π π 0 1 23 4 5 6 Steady-state probabilities: Balance Equations Flow into state 1-> Flow into 2 -> Flow into 3 -> <- flow out of state 1 Expected number of machines in the system 0.5 μ-λ 二0.167 Average (effective) arrival rate λ--λ 1 Expected waiting time in the queue(doesn't include service time) W- -0.167 W..9.60 90

The probability that there are no machines in the system (all servers are idle) is P,-[0.7143] c! (μ The probability of n machines in the queuing system is P (A)" P 0.2381 0 n- C Cl c The average number of machines in the queuing system is ајс し二 0.34286 ]

The average time a machine spends in the queuing system (waiting and being served) is W [0.3429] W.60.9 [185.143] min W:E The average number of machines in the queue is し,-[ 0.0095] The average time a machine spends in the queue, waiting to be served, is w,-(0.00952] Wy60-9=[5.143]