This question seeks to build and test your intuition on the design of waiting li
ID: 432304 • Letter: T
Question
This question seeks to build and test your intuition on the design of waiting lines with variability and two identical streams of customer arrivals (e.g., men and women). The three systems being considered are shown below, where the arrow denotes flow with arrival rate Rak, for stream k = 1, 2, the triangle represents the waiting line or queue and the rectangles represent processing by a server with specified average unit processing time – the two rectangles inside a larger rectangle in System B corresponds to a two-server station; System C is a single-server station with average unit process time cut by 50%. Assume values for Rak and p, for instance, take arrival rate Ra1 = Ra2 = 0.9 customers per hour, so total arrival rate in both systems B and C is Ra1 + Ra2 = 1.8 customers per hour and take p = 1 hour, so unit process time in System B is 1 hour and in System C is 0.5 hours. Unless otherwise stated, assume all coefficients of variation are 1.0. In the course material on waiting lines & the Manzana case, we saw that the independent system A has higher time in queue than the pooled system B. In parts III.A.1 & 2 below, we compare systems B and C.
Time in Queue, Tq: Calculate the average time in queue, Tq, in the Pooled System B and the Fast-server System C. Which system has the lowest average time in queue, Tq? Show your data values / approach.
Flow Time T comparison: Which system has the lowest flow time in system, T? Will the choice of which system has the lowest T continue to hold if the variability factor, V, is not 1 but much larger (e.g., V = (CVa2 + CVp2 )/2 = 3.5, corresponding to CVa2 = 2, and CVp2 = 5 or even larger V)? Briefly explain your Yes / No choice.
Qualitative Queuing Process Design: Consider the following three process designs to organize a call center with 12 employees. The center handles calls for two customer types. Type 1 customers call with credit card related questions and type 2 customers call with questions related to the online account opening. On a busy day the call center receives 60 calls per hour from type 1 customers and 30 calls per hour from type 2 customers. It takes on average 2 minutes to service both kinds of calls.
Process design 1: 6 employees handle type 1 calls; the other 6 employees handle type 2 calls
Process design 2: 8 employees handle type 1 calls; 4 employees type 2 calls.
Process design 3: 12 persons (cross-trained) handle all calls.
Which of the three process designs leads to the shortest (longest) average waiting time for a random incoming request? Why? (No quantitative calculations are necessary; only a cogent argument is.)
1 is the shortest, 2 is the longest
2 is the shortest, 1 is the longest
3 is the shortest, 2 is the longest
2 is the shortest, 3 is the longest
3 is the shortest, 1 is the longest
1 is the shortest, 3 is the longest
3 is the shortest, 1 and 2 are the same
Cannot be determined (explain why)
None of the above (explain why)
Ra1 Ra1 Ra2 Ra1 Ra2 72 Independent System (two single-servers) Pooled System B with two servers A Fast-server System C with one quicker serverExplanation / Answer
Formula for waiting time in Queue is as follows:
Parameters
System B
System C
Inter-arrival Time
a
1/0.85
= 1.18
1.18
Service Time
p
1.00
0.50
Number of Servers
m
2.00
1.00
Utilization
u = p/(a*m)
0.43
0.43
C.V. of arrival time
CVa
1
C.V. of servicing time
CVq
1
Waiting time in Queue
Tq
(0.59)(0.503)(1)
= 0.2960
(1.18)(0.74)(1)
= 0.87
Waiting Time in System (Flow time in system)
T = Tq + p
1.296
1.370
Lowest Tq is for system B, thus system B is better than System C, as it has lesser customers wait time.
The Lowest flow time, T is 1.296 for system B.
Adding variability in inter arrival and service time:
Parameters
System B
System C
Inter-arrival Time
a
1/0.85
= 1.18
1.18
Service Time
p
1.00
0.50
Number of Servers
m
2.00
1.00
Utilization
u = p/(a*m)
0.43
0.43
C.V. of arrival time
CVa
2
2
C.V. of servicing time
CVq
5
5
Waiting time in Queue
Tq
(0.59)(0.503)(14.50)
= 4.2915
(1.18)(0.74)(14.5)
= 12.16
Waiting Time in System (Flow time in system)
T = Tq + p
5.29
13.11
Lowest Tq is for system B, thus system B is better than System C, as it has lesser customers wait time.
The Lowest flow time, T is 1.296 for system B. Thus, the choice does not change if the variability is considered.
Formula for waiting time in Queue is as follows:
Parameters
System B
System C
Inter-arrival Time
a
1/0.85
= 1.18
1.18
Service Time
p
1.00
0.50
Number of Servers
m
2.00
1.00
Utilization
u = p/(a*m)
0.43
0.43
C.V. of arrival time
CVa
1
C.V. of servicing time
CVq
1
Waiting time in Queue
Tq
(0.59)(0.503)(1)
= 0.2960
(1.18)(0.74)(1)
= 0.87
Waiting Time in System (Flow time in system)
T = Tq + p
1.296
1.370
Lowest Tq is for system B, thus system B is better than System C, as it has lesser customers wait time.
The Lowest flow time, T is 1.296 for system B.
Adding variability in inter arrival and service time:
Parameters
System B
System C
Inter-arrival Time
a
1/0.85
= 1.18
1.18
Service Time
p
1.00
0.50
Number of Servers
m
2.00
1.00
Utilization
u = p/(a*m)
0.43
0.43
C.V. of arrival time
CVa
2
2
C.V. of servicing time
CVq
5
5
Waiting time in Queue
Tq
(0.59)(0.503)(14.50)
= 4.2915
(1.18)(0.74)(14.5)
= 12.16
Waiting Time in System (Flow time in system)
T = Tq + p
5.29
13.11
Lowest Tq is for system B, thus system B is better than System C, as it has lesser customers wait time.
The Lowest flow time, T is 1.296 for system B. Thus, the choice does not change if the variability is considered.
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