טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentZeltyn Sergey
SubjectCall Centers with Impatient Customers: Exact Analysis and
Many-server Asymptotics of the M/M/n+G Queue
DepartmentDepartment of Industrial Engineering and Management
Supervisor Professor Avishai Mandelbaum


Abstract

The subject of the present research is the M/M/n+G queue.  This queue is characterized by Poisson arrivals at rate λ, exponential service times at rate μ, n service agents and generally distributed patience times of customers. 

First, we provide an extensive background on the M/M/n+G model.  The following research is motivated by a phenomenon that has been observed in call center data: a clear linear relation between the probability to abandon P{Ab} and average waiting time E[W].   We analyze its robustness within the framework of the M/M/n+G queue, which gives rise to further theory and empirically-driven experiments .  

Then three asymptotic operational regimes for medium to large call centers are introduced and studied. These regimes correspond to the following three staffing rules, as λ and n increase indefinitely and μ held fixed :

Efficiency-Driven (ED):                             n ≈ (λ/μ) · (1-γ),  γ>0, 

Quality-Driven (QD):                                n ≈ (λ/μ) · (1+γ),   γ>0,   and

Quality and Efficiency Driven (QED):   n ≈ λ/μ + β (λ/μ)1/2,   -∞<β<∞.

In the ED regime, the probability to abandon and average wait converge to constants.  In the QD regime, we observe a very high performance level at the cost of possible overstaffing.  Finally , the QED regime carefully balances quality and efficiency: agents are highly utilized, but the probability to abandon and the average wait are small. 

Numerical experiments demonstrate that for a wide set of system parameters, the QED formulae provide excellent approximation for exact M/M/n+G performance measures. In turn, the much simpler ED approximations are very useful for overloaded queueing systems .   At the end, our theoretical results are applied to call-by-call data of a large bank.