Queueing M/M/s Calculator
Use our free Queueing mmscalculator Calculator to plan your operations & inventory strategy. Get detailed breakdowns, charts, and actionable insights.
Reviewed by Sahil, Senior Finance & Tax Editor
Formula
rho = lambda/(s*mu); Lq = (P0 * r^s * rho) / (s! * (1-rho)^2); Wq = Lq / lambda
Traffic intensity rho equals arrival rate divided by total service capacity. P0 is the probability of an empty system. Lq is the average queue length derived from P0, offered load, server count, and utilization. Wait time Wq follows from Little Law.
Worked Examples
Example 1: Call Center Staffing
Problem:A call center receives 20 calls per hour. Each agent handles calls at an average rate of 5 calls per hour. How many agents are needed to keep average wait under 2 minutes?
Solution:Lambda = 20 calls/hr, Mu = 5 calls/hr\nOffered load r = 20/5 = 4 Erlangs\nMinimum servers = 5 (rho must be < 1)\nWith s=5: rho = 20/(5x5) = 0.80, Wq = 0.1333 hr = 8 min (too high)\nWith s=6: rho = 20/(6x5) = 0.667, Wq = 0.0222 hr = 1.33 min (meets target)\nWith s=7: rho = 0.571, Wq = 0.0063 hr = 0.38 min
Result:6 agents needed. Wait time: 1.33 min | Utilization: 66.7%
Example 2: Hospital Emergency Room
Problem:An ER receives 12 patients per hour. Each doctor serves at a rate of 3 patients per hour. With 5 doctors, what are the performance metrics?
Solution:Lambda = 12, Mu = 3, s = 5\nOffered load r = 12/3 = 4 Erlangs\nUtilization rho = 12/(5x3) = 0.80\nP0 = 1.32%\nLq = 2.216 patients waiting\nWq = 0.1847 hr = 11.1 minutes\nWs = 0.5180 hr = 31.1 minutes\nPw = 65.2% chance of waiting
Result:Avg Wait: 11.1 min | Queue Length: 2.2 | Utilization: 80%
Frequently Asked Questions
What is the M/M/s queueing model and when is it used?
The M/M/s queueing model is a fundamental mathematical framework used in operations research to analyze waiting lines with multiple parallel servers. The first M stands for Markovian (memoryless) arrival process, meaning customers arrive according to a Poisson process with rate lambda. The second M indicates Markovian service times, meaning service durations follow an exponential distribution with rate mu. The lowercase s represents the number of identical parallel servers. This model is widely used in call centers to determine staffing levels, in hospitals to plan bed capacity, in banks to decide how many tellers to open, in computer networks to allocate processing resources, and in manufacturing to design workstation configurations. The model assumes infinite queue capacity and first-come-first-served discipline.
References
Reviewed by Sahil, Senior Finance & Tax Editor · Editorial policy