Two desks

What this queueing and exponential-distribution interview question tests

This is a hard probability question that combines queueing theory, exponential distributions, and conditional reasoning. It appears in the quantitative interview rounds at trading desks and asset-management firms, and it rewards both theoretical understanding and careful case analysis.

The problem hinges on recognising that you must account for which desk becomes free first, and then reasoning about the relative service times of the remaining two clients. Strong solutions set up the problem by conditioning on the order in which clients depart, leverage properties of the exponential distribution (particularly the memoryless property), and systematically enumerate the cases that contribute to your being last. The setup is more important than the arithmetic: candidates who can clearly state their assumptions and decompose the problem into tractable sub-cases typically receive credit even if final computation is messy.

• Competing exponential random variables and the minimum of independent exponentials
• The memoryless property and its consequences for conditional probability
• Case-by-case analysis and law of total probability