Question 1
A detailed analysis of the operating characteristics of the reservation system with one agent as proposed by the vice president of administration. What is your recommendation concerning a single-agent system?
The analysis shows that a single-agent system with one agent would not be a feasible solution as shown by Figure 1, 44% of the calls would be blocked answered, and 56% of the calls are answered. The airline wants a minimum of 85% of the calls answered and therefore this choice is not acceptable.
Figure.1 Waiting line model- 1 Agent with no waiting line
Mean ARRIVAL rate λ=1/3.75= .2667 custoers per minute or 16 customers per hour
Mean SERVICE rate μ=1/3 min〖= .3333 customers/minute or 20 customers per hour〗
Pj=((λ/μ)"^j)/j!/" ∑_(i=0)^k▒〖((λ/μ)"^i)" 〗/i!)
Sec. 11.8
K= # of servers
K=1
P1=((.2667/.3333)"^1)/1!/" ∑_(i=0)^1▒〖((.2667/.3333)"^0)" 〗/i!)
.8/(1+.8)=.44
P1= .44 x 100= 44%
Average number of units in waiting line, Lq
Lq means no queues allowed
Average number of units in system: L= (λ/μ)(1-P1) (11.32)
L= (.2667/.3333)(1-.44)= .448
Average time a unit spends in waiting, Wq = 0 not allowed
Average time a unit spends in system
W= Wq + (1/μ) Lq= L- (λ/μ) W= (Lq/ λ ) + (1/ μ) W= ((L)-( (λ/μ))/ λ)+(1/ μ)
((.448)-(.8)/(16)) + (1/20) = .028 x 60 min.= 1.68 min.
The current single agent will answer 44% of calls which is below the 85% company policy target.
Question 2
A detailed analysis of the operating characteristics of the reservation system based on your recommendation regarding the number of agents Regional should use.
Regional should use 2 agents with a block waiting system when comparing Figure 2, 3, and 4, as this would fulfill the requirement of 85% of calls answered, as well as keeping the cost down.
Figure 2. Waiting line model- 2 agents with no waiting (blocking)
Pj=((λ/μ)"^j)/j!/" ∑_(i=0)^k▒〖((λ/μ)"^i)" 〗/i!)
P2=((.2667/.3333)"^2)/2!/" ∑_(i=0)^2▒〖((.2667/.3333)"^i)" 〗/i!)
P2= (.32)/(1+.8+.32)= .15094
15% of calls will be blocked with 2 agents
Average customer in system: 2.54 minutes
L= (λ/μ)(1-P2)= (.8)(1-.15094)= .67924
W= Wq + (1/μ) Lq= L- (λ/μ) W= (Lq/ λ ) + (1/ μ) W= ((L)-( (λ/μ))/ λ)+(1/ μ)
((.67924)-(.2667/.3333))/.2667+(1/.3333)= 2.54minutes
Figure 3. Single Agent –waiting is allowed (M/M/1)
Mean ARRIVAL rate λ=1/3.75= .2667 custoers per minute or 16 customers per hour
Mean SERVICE rate μ=1/3 min〖= .3333 customers/minute or 20 customers per hour〗
P0= 1- (λ/μ) 1-(.8) = .20 20% probability that there are no customers in the system
Average number of customers in the waiting line
L= (λ )^2/( μ(μ- λ)) = (.2667)^2/( .3333(.3333- .2667)) = 3.2 customers in the waiting line
Average number in system L= Lq + λ/μ = 3.2 + (.2667/.3333) = 4.0002 customers
Average minutes a customer spends waiting (on hold)
Wq= Lq/ λ = 3.2/.2667 = 11.9985 minutes ~12 minutes waiting
Customer average minutes in system W= Wq + (1/μ) = 12 minutes + (1/.3333) = 15 minutes
Probability that a customer waits for service Pw= (λ/μ)=.2667/.3333= .8 80% chance of waiting
“n” units in system
Pn= (λ/μ)^n Po
Po= (.2667/.3333)0(.20) = .2000
P1= (.2667/.3333)1(.20) = .1600
P2= (.2667/.3333)2(.20) = .1280
P3= .1025
P4= .0819
Figure 4. Two Agents- waiting is allowed (M/M/2)
Po=1"/" ∑_(n=0)^(k-1)▒〖(λ/μ)^n/n!+(λ/μ)^k/k!(1/(1-λ/kμ))〗
According to standard numbers from Ch.11 (table 11.4)1
Ratio = .2667/.3333 = Po = .4286 42.86%
The probability that there are no customers in the system is 42.86%
Avg number of customers in waiting line
Lq=( (λ/μ)^k μ λ))/ (k-1)!(k μ- λ)2 x Po
Lq= (.2667/.3333)2(.2667)(.3333) / (2-1)!(2 x .3333 – .2667)2 x (.4286) = .1524
Average number of customers in the system
L= Lq + λ/μ = .1524 + (.2667/.3333) = .9526 ~ 1 customer per minute at all points in the system currently
Average minutes a customer spends waiting
Wq= Lq/λ = (.1524/.2667) = .571 min ~34 seconds
Average minutes a customer spends in the system
W= Wq + 1/ μ = .571 + (1/.3333) = 3.5713 minutes
Probability a customer waits for service
Pw= 1/k! (μ/λ)^k (k μ/ k μ-λ) = (.5)(.64)(1.67)(.4286) = .2290 22.90%
Probability “n” units in system k= 2
Pn= ((λ/μ)^n/n! Po.or Pn= ((λ/μ)^n//k!k(n-k)Po n>k k=2
Po= .4286
Pn1 = .3429
Pn2 = .1372
Pn3 = .0549
Pn4 = .0219
Question 3
A detailed analysis of the advantages or disadvantages of the expanded system. Discuss the number of waiting for callers the expanded system would need to accommodate.
The expanded system of 1 to 2 agents with call waiting, has the advantage of improving customer service relations through a more efficient call system that allows for almost no waiting. The disadvantage of this system is that it would create another cost to Regional airline of adding an additional agent and expanding the call system that is currently implemented. Furthermore, by analyzing tables 1-4, delineates the pros vs. cons of each system and the potential maximizing benefits from selecting 2 agents with call waiting. The 2-agent system with call waiting is the most efficient system, however it is costlier to Regional Airlines to add one more agent and change from blocked system to call waiting system.
According to figure 1-4 ‘s calculation, the purpose of this analysis is to individually assess the waiting line models individually to company policy as demonstrated on Table 1-4.
Table 1. Current company choice (fail company policy)
Individual Assessment of Waiting Line Model- 1 agent with no waiting line
Business Con Business Pro Customer Con Customer Pro
56% of customers at any moment do not get any service. Quick Service for the 44% that do get service If you are part of the 56% that gets turned away it will foster ill feelings If you are part of the 44%, you get speedy service
56% of possible lost revenue The majority will get a bad image of the company and reflect negatively on company review sites. If you are part of the 44% that got speedy service, you may become a repeat customer
Fail to reach 85% service company policy
Table 2. Second Choice (meets company policy)
Individual Assessment of Waiting Line Model- 2 agent with no waiting (block)
Business Con Business Pro Customer Con Customer Pro
15% of lost customers Reach company policy at 85% If you are part of the 15% that gets blocked, you will be angry 85% of customers only take 2.54 minutes of service
Additional $20/hr for second agent 2.54 minutes of service Negative feedback on reviews (more likely to leave negative reviews)
15% of business turned away
Table 3 (fail company policy)
Individual Assessment of Waiting Line Model- 1 agent where waiting is allowed (M/M/1)
Business Con Business Pro Customer Con Customer Pro
56% of customers at any moment do not get any service. Quick Service for the 44% that do get service If you are part of the 56% that gets turned away it will foster ill feelings If you are part of the 44%, you get speedy service
56% of possible lost revenue The majority will get a bad image of the company and reflect negatively on company review sites. If you are part of the 44% that got speedy service, you may become a repeat customer
Fail to reach 85% service company policy 80% chance of waiting for service
Table 4. Top Choice
Individual Assessment of Waiting Line Model- 2 agents where waiting is allowed (M/M/2)
Business Con Business Pro Customer Con Customer Pro
Pay for 2 agents Everybody gets served Wait for 34 seconds
(22.9% probability of waiting) Everyone gets service
Pay for waiting service Assume 1 customer per minute at all points in system Fast efficient service at 3.5713 min avg. customer in system
No one turned away, maxing out profitability No one turned away
Things to consider: Avg. booking airfare is ~$379.2 In essence, one additional customer would justify the addition of a second agent. In business, a return on investment mentality is crucial to justify expenditure as well as maximize profitability. It is justified to add call waiting and an additional agent as this will pay for itself and maximize profitability. This will also increase positive company perception to the customers by efficient service. A case could be made for 2 agents with no call waiting, but that also means that 15% of people are being blocked. Yes, that means that company policy is minimally met, but as proven by the addition of call waiting, the company can take care of all customers that call between 10-11am and have it pay off at the end to maximize profitability. In the psychology of waiting, people perceive a service/product worth obtaining as having to wait a certain amount of time.3 Much like people that perceive the word “free” as of less value, people that don’t wait for service also may not perceive the service as valuable or perceive the service/product as inferior to other competing factors.
Question 4
This report represents a pilot study of the reservation system for the 10:00 A.M to 11:00 A.M time period during which an average of one call arrives every 3.75 minutes; however, the arrival rate of incoming calls is expected to change from hour to hour. Describe how your waiting line analysis could be used to develop a ticket agent staffing plan that would enable the company to provide different levels of staffing for the ticket reservation system at different times during the day. Indicate the information that you would need to develop this staffing plan.
The analysis of the waiting line model could be useful for different times during the day. A study would need to be conducted to determine the average arrival rate and service time rates for the differing times. This would allow utilization of different waiting line models suitable to level of calls being made. If the calls are less than the current model with one agent and call blocking, the same system would be useful. If the call volume and frequency increase considerably then the data proves that the two-agent system should be favored for maximization. Future analysis could account for cost of the agent per hour vs. the cost of each booking. In addition, part time agent positions should be considered for cost control.
References
Anderson, David R. An Introduction to Management Science: Quantitative Approaches to Decision Making. Cengage Learning, 2016.
Meister, David H. “Articles.” David's Resources, davidmaister.com/articles/the-psychology-of-waiting-lines/.
Williams, Geoff. “5 Creative Ways to Cut Airfare Costs.” U.S. News & World Report, U.S. News & World Report, 13 Sept. 2013, money.usnews.com/money/personal-finance/articles/2013/09/18/5-creative-ways-to-cut-airfare-costs.