Interactive Queueing Theory: Real-Time Analysis of M/M/c Systems

Research Team
1 Research Platform
Abstract —

This paper demonstrates the interactive capabilities of the Research platform by exploring M/M/c queueing systems with a live calculator. Readers can adjust parameters in real-time to understand how arrival rates, service rates, and server counts affect queue performance.

Keywords: queueing theory, M/M/c queue, Erlang C, operations research, interactive visualization

Introduction

Queueing theory provides mathematical models to analyze waiting lines and service systems. The M/M/c queue is a fundamental model where:

  • M (Markov): Arrivals follow a Poisson process with rate λ\lambda
  • M (Markov): Service times are exponentially distributed with rate μ\mu
  • c: Number of parallel servers

This model is widely used in call center staffing, hospital emergency room capacity planning, computer server resource allocation, and retail checkout lane optimization.

Mathematical Foundation

For an M/M/c queue to be stable, the traffic intensity per server must be less than 1:

ρ=λc⋅μ<1\rho = \frac{\lambda}{c \cdot \mu} < 1

where λ\lambda is the arrival rate (customers per unit time), μ\mu is the service rate per server, and cc is the number of servers.

Key Performance Metrics

The probability that all servers are busy (Erlang C formula) is given by:

C(c,u)=ucc!⋅11−ρ∑k=0c−1ukk!+ucc!⋅11−ρC(c, u) = \frac{\frac{u^c}{c!} \cdot \frac{1}{1-\rho}}{\sum_{k=0}^{c-1} \frac{u^k}{k!} + \frac{u^c}{c!} \cdot \frac{1}{1-\rho}}

where u=λ/μu = \lambda/\mu is the offered load.

The mean waiting time in the queue is:

Wq=C(c,u)μ⋅c⋅(1−ρ)W_q = \frac{C(c, u)}{\mu \cdot c \cdot (1 - \rho)}

The mean queue length follows from Little’s Law:

Lq=λ⋅WqL_q = \lambda \cdot W_q

Interactive Calculator

Use the calculator below to explore how different parameters affect queue performance. Adjust the arrival rate λ\lambda, service rate μ\mu, and number of servers cc to see real-time changes in the metrics.

Adjust Parameters

Queue Performance Metrics

Parameters: λ = 20, μ = 1.5, c = 15

Understanding the Metrics

The calculator displays four key performance indicators:

  1. Utilisation (ρ\rho): The proportion of time servers are busy. Higher utilisation means more efficient use of resources, but values approaching 100% lead to long queues.

  2. Probability of Waiting P(W>0)P(W > 0): The likelihood that an arriving customer must wait. This equals C(c,u)C(c, u), the Erlang C formula.

  3. Mean Wait Time (WqW_q): The expected time a customer spends waiting in the queue before service begins.

  4. Queue Length (LqL_q): The average number of customers waiting in the queue (not including those being served).

Practical Applications

Call Center Example

Consider a call center with:

  • 30 calls per hour (λ=30\lambda = 30)
  • Each agent handles 4 calls per hour on average (μ=4\mu = 4)
  • 10 agents available (c=10c = 10)

The utilisation would be:

ρ=3010×4=0.75=75%\rho = \frac{30}{10 \times 4} = 0.75 = 75\%

Use the calculator above to experiment with these values and see how adding or removing agents affects wait times.

Conclusion

Interactive calculators like the one in this paper enable readers to develop intuition for mathematical models by experimenting with parameters in real-time. This approach bridges the gap between theoretical equations and practical understanding.

References

  1. Erlang, A. K. (1909). “The Theory of Probabilities and Telephone Conversations”. Nyt Tidsskrift for Matematik.

  2. Kleinrock, L. (1975). Queueing Systems, Volume 1: Theory. Wiley-Interscience.

  3. Gross, D., Shortle, J. F., Thompson, J. M., & Harris, C. M. (2008). Fundamentals of Queueing Theory. Wiley.