Operations 311 Management Fall 2008 Solution: Homework # 3 Managing Uncertainty 1. (30 points, 6 points each) Chapter 8A Question 7 (page 301 in the textbook). a. M/D/1 system = 10 person/hour, Ca = 1 S = 1, = 12 person/hour, Cs = 0 = 5/6 = 83% 2 2( S +1) C a + C s2 (5 / 6) 2 1 25 Lq = = = = 2.08 people 1 2 1/ 6 2 12 b. Ls = Lq + S = 2.92 people c. Wq = Lq/ = 0.208 hour d. Ws = Wq + 1/ = 0.292 hour e. = 12 person/hour, Lq becomes infinity 2. (20 points, 5 points each) Chapter 8A Question 16 (page 303 in the textbook). a. M/M/1 system = 6 person/hour, = 10 person/hour, Ls = / ( ) = 1.5 people Ws = Ls/ = 0.25 hour = 15 minutes If the students use a different equation (the general approximation this is fine) b. = / = 6/10 = 60% c. P0 = 1 / = 40%, P1 = (1 / ) ( /) = 24%, P2 = (1 / ) ( /)2 = 14.4%, P(N>2) = 1 (P0 + P1 + P2) = 21.6% d. M/M/2 system Macro gives us Lq = 0.0593 Wq = Lq/ = 0.0099 hour Ws = Wq + 1/ = 0.1099 hour Or 1
= 6 person/hour, Ca = 1 S = 2, = 10 person/hour, Cs = 1 = 6/(2*10) = 30% 2 2( S +1) C a + C s2 0.3 6 Lq = = = 0.0748 people 1 2 0.7 Wq = Lq/ = 0.0125 hour Ws = Wq + 1/ = 0.1125 hour 3. (20 points, 7 points for a & b, 6 points for c) The Northern Emergency Computer Support has a call center with 10 customer service representatives. Historical statistics shows that customer calling times are exponentially distributed, and are 20 minutes on average. Customers arrive according to a Poisson process having rate 18 customers per hour. Using approximation formula for your calculation. a. How long on average does a customer have to wait? M/M/10 system = 18 person/hour, Ca = 1 S = 10, = 3 person/hour, Cs = 1 = 18/30 = 0.6 2 2( S +1) C a + C s2 Lq = = 0.228 people 1 2 Wq = Lq/ = 0.0127 hour = 0.759 minute b. The Northern Emergency Computer Support tries to merge with the Southern Emergency Computer Support who also has a call center with 10 customer service representatives and faces exact the same customer demand pattern. While the managers want to maintain the same waiting time criteria --- average waiting time to be less than 1 minute, a consultant points out that after the merge, the company in fact can fire 3 representatives and retain only 17 customer service representatives. Is the consultant correct? = 36 person/hour, Ca = 1 S = 17, = 3 person/hour, Cs = 1 = 36/51 = 0.706 2 2( S +1) C a + C s2 Lq = = 0.421 people 1 2 Wq = Lq/ = 0.0117 hour = 0.701 minute The consultant is correct. c. Define risk pooling. What is the risk pooling effect in this question? Risk pooling definition: By aggregate individual processes, the variability is reduced and better performance can be achieved.
2
In the example, risk pooling enables better performance with a smaller work force. Be flexible here. 4. (30 points, 10 points each part) You are working at a bank and doing resource requirements planning. You think that there should be six tellers working in the bank. Teller service time takes in average 15 minutes per customer with a standard deviation of 5 minutes. On average one customer arrives in every three minutes according to an exponential distribution. a. On average how many customer would be waiting in line? M/G/6 system = 20 person/hour, Ca = 1 S = 6, = 4 person/hour, Cs = 5/15 = 0.33 = 20/(6*4) = 83% 2 2 ( S +1) C a + C s2 Lq = = 1.68 people 1 2 This is one possible answer. The majority will ignore the 5 minuets of SD and will do it in the following way:
2( S +1) C a2 + C s2 0.83 2( S +1)) 1 + 1 Lq = = = 2.92 1 2 1 0.83 2
b. On average how many customer would be in the bank? Ls = Lq + S = 1.68 + 5 = 6.68 people Clearly we need to adjust here. This will also be correct: Ls = Lq + S = 1.68 + 5 = 2.92 + 5 = 7.92 people c. On average how long would a customer spend in the bank? Ws = Ls/ = 0.334 hour = 20.05 minutes Similarly this will also be correct Ws = Ls/ = 7.92/20 =0.39 hour = 24 minutes
3