Tools › Operations Research › Queueing and Simulation › Queueing and Capacity › Example
Hillier and Lieberman: County Hospital emergency room, one doctor (M/M/1) against two (M/M/2)
| Tool | Queueing and Capacity |
| Source | Hillier, F. S., and Lieberman, G. J. (2015). Introduction to Operations Research, 10th ed. McGraw-Hill, New York. Chapter 17, the County Hospital emergency room example (M/M/1 and M/M/s). |
| Level | Undergraduate |
Problem
Patients arrive at a hospital emergency room at a mean rate of 2 per hour and one doctor treats them at a mean rate of 3 per hour, both exponential. Find the utilization, the mean number of patients waiting and in the room, the mean waiting and sojourn times and the probability a patient waits more than an hour; then repeat with a second doctor on duty.
Inputs to Enter
- Model M/M/1, rate basis per hour, arrival rate λ = 2
- Service input 'Rate per server', μ = 3; wait time t = 1 h for the tail probability
- Then model M/M/c with c = 2 servers, same rates
Answers
- M/M/1: ρ = 2/3, P0 = 1/3, L = 2, Lq = 4/3, W = 1 h, Wq = 2/3 h (40 min), P(W > 1 h) = e⁻¹ = 0.3679, P(Wq > 1 h) = ρ e⁻¹ = 0.2453
- M/M/2: ρ = 1/3, P0 = 0.5, Lq = 1/12, L = 0.75, Wq = 1/24 h (2.5 min), W = 0.375 h, P(wait) = 1/6
- Little's law L = λ W holds in both cases
Notes
Not the default case (the default is the tool's Erlang C call-centre case of 360 calls/h, 180 s handle time, 20 agents, which has no printed source and is checked against an independent direct-sum computation). Choose the model in the Model select and switch the service input to 'Rate per server'.