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OPM 661 - Business Analytics

创建日期：2024-04-22 17:18:19

所属分类：商业分析business analysis

标签： OPM 661

Business Analytics

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OPM 661 - Business Analytics: Robust Planning in Stochastic Systems

Assignment II: Performance evaluation of service systems with abandonments
The hotline of a university experiences high customer waiting times or overstaffing on a
regular basis. The call center’s analyses show that the number of employees working is
often not aligned with the demand for the service, i.e., the call volume. This leads to a high
number of customers abandoning while staying in the queue. In order to increase customer
satisfaction, the call center wants to understand how such a queue behaves by using methods
from Business analytics. The goal is to learn about the sensitivity of performance measures
with respect to the number of servers (staffing level) and in the abandonment behaviour.
Orbit

Figure 1: Queueing model of the call center with 7 agents
For a certain period, the call center should be analyzed under steady-state assumptions. Cus-
tomers arrive according to a Poisson process with an arrival rate of λ customers per hour.
Due to high data analytics skills, arrival rates can be accurately forecasted by the call center.
It is also known, that the service times of the call center agents are exponentially distributed
with rate of µ customers per hour. Customers that are waiting in the queue abandon after an
exponentially distributed waiting time with rate ν > 0. Abandoned customers reach the so
called orbit (not under control of the call center) with probability p and leave the system with
probability 1 − p. Customers in the orbit call again after an exponentially distributed time
with rate γ.
The call center has a total of K telephone lines, that means at most K customers can be
present in the system (being served or waiting in the queue). If all K lines are occupied,
additional customers are blocked. The call center can be described as a piecewise station-
ary queueing system with abandonments and retrials as depicted in Figure 1. The orbit is
assumed to be limited to the arbitrary high number of J customers.
1
Tasks:
Derive a Markov chain that describes the number of customers in the system for a given
number of employees c. Derive formulas for the expected performance measures dependent
on the steady-state probabilities. Implement the Chapman-Kolmogorov equations (CKEs)
in Python. Finally, perform a sensitivity analysis to generate insights about the effects of
abandonment rate on the performance measures.
a) Describe the state space representation. (0.5 P.)
b) Draw the general1 state transition diagram for the continuous-time Markov chain and an
arbitrary number of employees c. Derive the Chapman-Kolmogorov equations (CKEs).
(3 P.)
c) Assume the steady-state probabilities Ps for all states s ∈ S are known. Provide the
formulas to calculate the expected number of customers in the call center E[LS], the
expected queue length E[LQ], the expected number of clients in the orbit E[LO], and the
expected utilization of the call center E[U ]. (1 P.)
d) The call center estimates the arrival rate λ = 30, abandonment rate ν = 10, retrial
probability p = 0.5, and the retrial rate γ = 5 customers per hour, while the service rate
is µ = 5. Solve the CKEs and calculate steady-state probabilities using Python for the
estimated rates, an orbit capacity of J = 100 customers, K = 10 telephone lines and
c = 7 agents. (2 P.)
e) Visualize the calculated steady-state probabilities with an appropriate visualization tool.
(0.5 P.)
f) Calculate the performance measures from task c) utilizing the values for steady-state
probabilities. (1 P.)
g) Perform a sensitivity analysis of the impact of ν on the performance measures for several
number of agents. Discuss your findings. (2 P.)
General remarks:
• Solve the assignment in groups of 3 to 4 students (same groups as for assignment I).
Submissions of groups with fewer or more students will not be accepted.
• If in doubt about specific assumptions, make reasonable assumptions on your own and
clearly state them in your report.
Enjoy the assignment!
1without utilizing values for parameters (algebraic)

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