Scenario: A bank wants to choose between a single‑line multi‑server queue and multiple independent queues.
Question 1: Compare expected waiting time and variance between a single‑line multi‑server system and multiple independent queues.
Question 2: Under what customer‑load scenarios does one system outperform the other?
Question 3: How would you model customer experience and throughput analytically?
Question 4: What data would you collect post‑implementation to validate the design choice?
Quick Answer: Evaluates queueing-system reasoning for single-line versus multiple-line bank teller designs. Strong answers compare M/M/c pooling with parallel M/M/1 queues, waiting-time averages, variance, stability, fairness, and operations.
Bank Branch Queue Design: Single Queue Versus Multiple Queues
A bank branch has c identical tellers. Customers arrive approximately as a Poisson process at rate lambda, and each teller has independent exponential service times with rate mu. Compare two designs:
A single serpentine line feeding the first available teller, an M/M/c system.
c independent lines, one per teller, where customers choose a line uniformly at random, approximated as c parallel M/M/1 queues each with arrival rate lambda/c.
Constraints & Assumptions
Assume first-come, first-served service and no balking or reneging initially.
Compare both average waiting time and variability.
State the stability condition.
Discuss practical deviations such as shortest-line choice, customer psychology, and fairness.
Clarifying Questions to Ask Guidance
Are tellers identical, and can any teller serve any customer?
Do customers choose lines randomly, shortest-line, or based on perceived speed?
Does the branch optimize average wait, tail wait, fairness, or customer satisfaction?
Are there priority customers or specialized service types?
Part 1 - Expected Waiting and Variance
Compare expected waiting time and waiting-time variance for the two designs.
What This Part Should Cover Guidance
Define utilization rho = lambda / (c mu).
For a single queue, discuss pooling and Erlang C for M/M/c.
For multiple random queues, describe each line as M/M/1 with arrival rate lambda/c.
Explain why pooling generally reduces average wait and variability.