In the Cachon-Terwiesch textbook, there is an approximation formula for the aver
ID: 435085 • Letter: I
Question
In the Cachon-Terwiesch textbook, there is an approximation formula for the average waiting time, Tq, in a G/G/m queue with m identical servers:
1. Based on the Tq approximation formula, compare a single-server (m = 1) waiting line system to a two-server (m = 2) system with the same average customer inter-arrival time “a”, server utilization, u, and coefficients of variation (CVs with squared CVs for inter-arrival and service times denoted by CVa2 and CVp2respectively). Let Tq, represent average waiting time in queue, T be the system (=queue + servers) flow time, and let Ip, Iq, and I represent the average inventory at the servers, in the queue, and system, respectively. Which of the following 10 statements are true (pick ALL that apply)?
a. The two-server system always will be better than the single-server system in terms of the four waiting line metrics, Tq, T, Iq, and I.
b. The single-server system will be busier on average than the two-server system.
c. Both systems will satisfy I = (1/a) T, Iq = (1/ a) Tq.
d. Average number of customers in the waiting line Iq will be the same in both systems.
e. The directional (greater than, smaller than) comparison between Tq in the two systems will depend on the value of variability factor, V=(CVa2+CVp2)/2 for V > 0.
f. The directional comparison between T in the two systems will depend on the value of variability factor, V=(CVa2+CVp2)/2, for V > 0
g. If the variability factor V is 1 (e.g., all CVs are 1), the two-server system will likely have lower Iq.
h. If the variability factor is 1 (e.g., all CVs are 1), the single-server system will likely have lower I.
i. If the variability factor is 1 (e.g., all CVs are 1), then T will be higher in the two-server system, but Tq will be higher in the single-server system.
j. The two-server system will have higher average number of customers in process being served, Ip, than the single-server system.
utilizationV2(m) -1 1-utilization Activity time Time in queue T. -Explanation / Answer
Based on the Tq approximation formula, compare a single-server (m = 1) waiting line system to a two-server (m = 2) system with the same average customer inter-arrival time “a”, server utilization, u, and coefficients of variation (CVs with squared CVs for inter-arrival and service times denoted by CVa2 and CVp2respectively). Let Tq, represent average waiting time in queue, T be the system (=queue + servers) flow time, and let Ip, Iq, and I represent the average inventory at the servers, in the queue, and system, respectively
Solution
The following statements are true :-
a. The two-server system always will be better than the single-server system in terms of the four waiting line metrics, Tq, T, Iq, and I.
Reason - As the multi channel or two server system will have higher rate of service as at a given point of time when a single server is servicing one customer, two-server will be servicing two customers. Thus waiting time for customers will be less and more customers can be serviced within the same span of time compared to a single-server system.
b. The single-server system will be busier on average than the two-server system.
Reason - A single server will be accumulating a longer queue as compared to two-server system for a given period of time, hence it will be busier.
e. The directional (greater than, smaller than) comparison between Tq in the two systems will depend on the value of variability factor, V=(CVa2+CVp2)/2 for V > 0.
Reason - Variability is something that causes anomaly in the system and deviate from its predictable behaviour. Hence, if there is some variability in the system, the Tq of the system will also be affected.
f. The directional comparison between T in the two systems will depend on the value of variability factor, V=(CVa2+CVp2)/2, for V > 0
Reason - Variability is something that causes anomaly in the system and deviate from its predictable behaviour. Hence, if there is some variability in the system, the T of the system will also be affected too.
j. The two-server system will have higher average number of customers in process being served, Ip, than the single-server system.
Reason - In a multi server system servicing rate is higher as compared to a single server system as there arep more channels and at a given point of time more number of customers are served than the single channel system.
g. If the variability factor V is 1 (e.g., all CVs are 1), the two-server system will likely have lower Iq.
Reason - lq represents the average inventory at the queue, with variability factor being constant, servicing rate will be higher in two-server system and thus inventory depriciation will be at a faster rate, resulting to lower lq.
i. If the variability factor is 1 (e.g., all CVs are 1), then T will be higher in the two-server system, but Tq will be higher in the single-server system.
Reason - As variability factor being constant , the system flow time will be at a faster rate and hence will be higher in the two-server sytem. Whereas the average waiting time in the queue will be more in the single server system, as average rate of servicing will be slower.