M/M/1 Queuing Calculations

Introduction

The purpose of this document is to provide an overview of M/M/1 queuing systems, and in particular calculations involved with those systems. A few examples and R code will demonstrate some of the calculations. M/M/1 queues, and queuing systems in general are also easily studied with simulation. The Georgia Tech course Simulation Analysis (ISYE 6644) goes over queuing systems, and also how to construct simulations of these systems using Arena. As queuing systems become more complex, the ability to analyze them using “closed form” or numerical solutions becomes more difficult. Simulation techniques provide a way to tackle more complex problems. This document will focus only on M/1/1 queues, which are among the most basic forms of queuing systems studied in operations research.

The contents included here are NOT part of a homework assignment, or exam, but rather just a “re-working” of the examples used in the course content. Georgia Tech is very much concerned about academic integrity and honesty. My goal is not to cheat, but rather the help others understand the lecture notes.

The original work was done by Bob Myers, Lecturer, Georgia Institute of Technology, MGMT-6203 Data Analytics for Business, Summer 2020.

M/M/1 Queues

M/M/1 Queues can be easily described using a diagram:

MM1 Queue - Georgia Tech MGMT 6203

There are several assumptions about M/M/1 queues:

• There is in infinite supply (population) of customers
• The arrivals are random, and follow a Poisson distribution with rate \(\lambda\)
• The size of the queue can become infinite
• There is a single queue for customers to wait in
• Customers are processed in FIFO (First come, first served)
• Customers don’t “balk” or leave the queue once they enter and prior to being served
• There is a single “channel” or service phase with random service times that follow an Exponential distribution with mean service rate of \(\mu\)
• The system is a single phase. (1 queue, one service point.)

Formulas

Given arrival rate=\(\lambda\) and service times=\(\mu\), the following metrics can be computed for a given M/M/1 system in closed (numerical) form, and without requiring simulation:

Utilization: \(\rho=\frac{\lambda}{\mu}\)

Average num. of customers in the system: \(L_s=\frac{\lambda}{\mu-\lambda}\)

Average num. of customers in the queue: \(L_q=\frac{\lambda^2}{\mu(\mu-\lambda)}=L_s*\rho\)

Average time a customer spends in the system: \(W_s=\frac{1}{\mu-\lambda}=\frac{L_s}{\lambda}\)

Average time a customer spends in the queue: \(W_q=\frac{\lambda}{\mu(\mu-\lambda)}=\frac{L_q}{\lambda}\)

Probability of n units in the system: \(P_n=(1-\frac{\lambda}{\mu})(\frac{\lambda}{\mu})=(1-\rho)(\rho)^n\)

The utilization for X% service level: \(1-\rho^n=X\)%

Example Calculations in R

The following R code will calculate the above metrics based on the given values of \(\lambda=15\), \(\mu=20\), and \(n=3\):

rm(list=ls())

mm1_metrics <- function(lambda, mu, n, sl){
  rho <- lambda/mu
  Ls <- lambda/(mu-lambda)
  Lq <- lambda**2/(mu*(mu-lambda))
  Ws <- 1 / (mu-lambda)
  Wq <- Lq/lambda
  Pn <- (1-rho)*(rho)**n
  X <- data.frame(rho, Ls, Lq, Ws, Wq, Pn)

  names(X)<-c('rho','Ls','Lq','Ws','Wq','Pn')
  return(X)
}

mm1_metrics(15, 20, 3)

##    rho Ls   Lq  Ws   Wq        Pn
## 1 0.75  3 2.25 0.2 0.15 0.1054688

Note that if the above inputs are in hours, then the Ws and Wq values will also be in hours. We would need to multiply these by 60 to get the number of minutes.

Probability of a Max N in System

Consider the following:

Because of limited space, the banker would like to ensure with a 95% confidence (or service level) that no more than 3 cars will be in the system at any time. What confidence exists currently for no more than 3 cars? What rate would the teller need to operate at in order to meet a 95% service level? What would be the utilization at this point?

The following R code will compute the probability of a maximum of n customers in the system given an arrival rate of \(\lambda=15\) and an average service rate of \(\mu=20\).

prob_of_max_n_in_system <- function(lambda, mu, maxn) {
  p=0
  
  for(n in seq(0,maxn)){
    px <- mm1_metrics(lambda, mu, n)
    p = p + px$Pn 
  }
  return(p)
}

prob_of_max_n_in_system(15,20,3)

## [1] 0.6835938

To answer the second part of the question, namely what rate would the teller need to operate at in order to meet a 95% service level and what would the utilization be at this point, we have the following:

required_rate_and_utilization <- function(lambda, n, p){
  r = (1-p)**(1/(n+1))
  m = lambda / r
  X = data.frame(r, m)
  names(X) <- c('rho','mu')
  return(X)
}

required_rate_and_utilization(15, 3, .95)

##         rho       mu
## 1 0.4728708 31.72114

This concludes this write-up on how to do M/M/1 Queue calculations.