Queue Design and Economics, Explained with Examples

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Queue Design and Economics

Quantitative Analysis · Chapter 4: Inventory Management and QueuingSection 4.8 · lesson 19 of 5511 min · free with a free account

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In this lesson

How waiting-line theory lets a manager predict wait times, queue lengths and utilization before a single server is hired.

  • Define traffic intensity and explain why it must stay below 1
  • Apply the M/M/1 formulas and Little's Law to any stable queue
  • Extend the analysis to multi-server M/M/s systems
  • Find the cost-minimizing number of servers, and test how robust it is

Transcript

Everything said in the video, section by section (1,665 words). Read along, or search it.

Inventory Management and Queuing · Section 4.8

Queue Design and Economics

Welcome to section four point eight, Queue Design and Economics. Every business that serves customers in person faces the same tension. Banks, hospitals, call centres, drive-throughs, airline check-ins. Add more servers and customers wait less, but capacity sits idle. Run lean, and capacity is fully used, but customers grow restless. Some of them simply leave, and you never find out why. Queuing theory turns that trade-off into a small set of formulas. With them you can predict wait times, queue lengths and utilisation. And you can do it before a single server is hired. We will build the single-server formulas, then extend them to many servers. Then we attach money to the answer, and staff a real bank branch.

Two Parameters

The Numbers That Drive Every Queue

Every waiting-line analysis begins with exactly two numbers. The first is the arrival rate, written lambda. It measures how fast customers enter the system, usually per hour. The second is the service rate, written mu. It measures how fast one server processes them, in the same units. Divide arrivals by service and you get traffic intensity, called rho. Rho is also called utilisation, the fraction of time the server is busy. If rho is below one, service outpaces arrivals, and the queue is stable. If rho reaches one or more, arrivals win in the long run. The queue then grows without bound, and no formula will save you. So check stability first. Everything else assumes it.

Kendall Notation

What M/M/1 Actually Says

The label M M one is not arbitrary shorthand. It follows a convention introduced by the mathematician David Kendall. The first M means arrivals follow a Markovian, or Poisson, process. That means random and memoryless. The next arrival does not care how long ago the last one came. The second M means service times are exponentially distributed, also memoryless. A job half finished tells you nothing about how much is left. And the one simply means a single server. So the system is a straight line of four stages. Random arrivals, one first-in first-out queue, one server, then departure. The notation also extends. M G one relaxes service times to a general distribution. Naming these assumptions matters, because the M M one formulas hold only while all three of them do.

Core Formulas

The Four M/M/1 Measures

Once rho is below one, four measures describe the system completely. Two of them count customers. Two of them measure time. L sub s is the average number in the system, lambda over mu minus lambda. L sub q counts only those waiting, lambda squared over mu times mu minus lambda. W sub s is the average time from arrival to departure, one over mu minus lambda. And W sub q is the wait before service even begins. Now notice what all four share. The term mu minus lambda sits in every single denominator. As arrivals creep up toward the service rate, that term shrinks toward zero. And every measure shoots toward infinity. That is the hockey stick. So ninety per cent utilisation looks efficient. In fact it is already a crisis. At rho of zero point nine, the average customer waits about nine times longer than at zero point five. That is why good service businesses hold rho below zero point eight. The slack is not idle waste. Customers experience it as quick service.

Little's Law

One Identity That Holds Everywhere

Beneath those four formulas sits something deeper and far more general. It is called Little's Law, after John Little, who proved it in nineteen sixty-one. It is remarkable because it holds for any stable queueing system. Not just M M one. Not just Markovian. Not just single server. The one condition it does need is that the system has reached steady state. It says L equals lambda times W. The average number in the system equals arrivals per hour, times average time in the system. It works on the queue alone, and on the whole system. It also works on factories, hospitals, software pipelines and supply chains. Its practical value is simple. Measure one quantity, then compute the other. A coffee shop counts fifteen customers inside on average, with one hundred and twenty entering per hour. Fifteen over one hundred and twenty is zero point one two five of an hour, about seven and a half minutes.

Multi-Channel

M/M/s - Many Servers, One Line

Most real service systems have more than one server. A bank has several tellers. A hospital has several doctors. A call centre has dozens of agents on the floor. The M M s model extends M M one to s parallel servers. They share a single queue, and each works at the same rate, mu. So arrivals join one shared line, not a separate line per server. Whoever is at the front takes whichever server frees up first. Total capacity is therefore s times mu. And the stability condition becomes lambda over s times mu, below one. Now the counter-intuitive part. Adding servers does not simply divide the wait by s. Any free server can take any customer, so the first extra helps enormously, and each one after it helps less.

The Bulky Part

The M/M/s Formulas

The M M s formulas are bulkier than M M one, for one specific reason. P nought is the probability that the system is completely empty. It cannot be written as a single closed expression. It needs a summation over n, from zero up to s minus one. Then a second term carrying a factorial in its denominator. Once P nought is known, the queue length L q follows directly. One warning on notation here. In these formulas rho means lambda over mu, not the per-server utilisation. So rho over s is the utilisation of each individual server. In practice nobody computes this by hand beyond two or three servers. Spreadsheets and queueing software handle the factorials and the summations for you.

The Economics

Which Number of Servers Is Best?

The formulas tell you what happens at each level of s. They do not tell you which s is best. That is economics, not algebra. Every server you add cuts waiting cost. Less time lost, fewer walkouts, better satisfaction. Every server you add also raises service cost. Wages, equipment, training. So total cost is C w times L q, plus C s times s. C w is the customer waiting cost per unit of time. Often dollars per customer per hour, reflecting lost sales and satisfaction impact. C s is the cost per server per unit of time, wages plus overhead. The rule is mechanical. Evaluate the total at one server, then two, then three. Choose the smallest number where adding one more would raise total cost, not cut it.

Worked Example

Bank Branch: The Single-Teller Case

Now put all of it to work on a real bank branch. At peak the branch records eighteen arrivals per hour, and each teller serves twenty. Waiting cost is estimated at twenty-five dollars per customer per hour. That figure reflects walkouts and reputational damage, not a ticket price. Each teller costs fifteen dollars an hour, wages plus overhead. Start with one teller, and check stability first. Rho is eighteen over twenty, so zero point nine. Stable, but only just. The average number in the system is eighteen over two, so nine customers. The average number waiting is three hundred and twenty-four over forty, so eight point one. Time in the system is half an hour, and time in the queue is twenty-seven minutes. So twenty-seven minutes queuing, then three minutes of actual service.

Second Teller

The M/M/2 Calculation

Add a second teller and everything changes. Total capacity becomes two times twenty, so forty customers per hour. Utilisation per server drops to eighteen over forty, zero point four five. That is well clear of the danger zone. Now the arithmetic. First, P nought. The summation is one plus zero point nine, giving one point nine. The last term is zero point eight one over one point one, about zero point seven four. Add them and invert. P nought is about zero point three eight. Feed that into the L q formula and the queue is zero point two three customers. Little's Law converts that to a wait, about zero point zero one three of an hour. That is under a minute. The queue fell from eight point one to under a quarter of a customer. A ninety-seven per cent reduction, bought for fifteen dollars an hour.

The Decision

Costing Every Configuration

Queue length is only half the answer. Now attach the money. Waiting cost is twenty-five dollars times L q, every hour. Service cost is fifteen dollars times the number of tellers. With one teller, waiting cost alone is two hundred and two dollars fifty. Add the wages and the hourly total is two hundred and seventeen dollars fifty. With two tellers, waiting cost collapses to five dollars seventy-one. Wages double to thirty dollars, so the total is thirty-five dollars seventy-one. A third teller cuts waiting cost to seventy-five cents. But wages rise to forty-five dollars, so the total climbs to forty-five dollars seventy-five. It only gets worse from there. Four tellers cost sixty dollars ten in total. So the optimum is two tellers, at about thirty-five dollars seventy-one an hour. That move saves one hundred and eighty-one dollars seventy-nine, every single hour. A third teller fails because the extra waiting cost it saves is smaller than its wage.

Robustness

Sensitivity - Does the Answer Hold?

One optimal answer can mislead a manager who treats it as fixed. The real value of the model is that you can re-run it. Change arrivals, service rate, waiting cost, or server cost, and look again. Suppose peak arrivals climb from eighteen to twenty-five per hour. Utilisation at two tellers rises to twenty-five over forty, zero point six two five. The queue lengthens, and eventually a third teller becomes justifiable. Or suppose management decides a frustrated customer costs fifty dollars an hour. Perhaps because of social-media risk. Waiting then weighs more heavily. If server cost falls instead, through cross-training branch staff, more servers are warranted. Off peak, with fewer arrivals, the optimum may drop back to a single teller. Which is why many branches add extra tellers only during peak windows. For this bank the answer is robust. Even at fifty dollars, two tellers still wins.

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FAQ

Questions students ask us

About Queue Design and Economics

What is traffic intensity in queuing theory, and why must it stay below 1?
Traffic intensity, or utilization, is \rho = \lambda / \mu, the arrival rate divided by the service rate, and it shows the share of time a server is busy. If \rho is 1 or more, arrivals outpace service and the queue grows without bound. Waits rise steeply as \rho nears 1: in M/M/1, the wait at \rho = 0.9 is about nine times the wait at 0.5.
What are the M/M/1 queuing formulas?
For a stable single-server queue, L_s = \lambda/(\mu - \lambda), L_q = \lambda^2/[\mu(\mu - \lambda)], W_s = 1/(\mu - \lambda) and W_q = L_q/\lambda. With 18 arrivals and a service rate of 20 per hour, there are 9 customers in the system, 8.1 waiting, 30 minutes in the system and 27 minutes in the queue. Every formula depends on \mu - \lambda, so measures explode as it shrinks.
What is Little's Law?
Little's Law, proved by John Little in 1961, states L = \lambda W: the average number in a system equals the arrival rate times the average time spent there. It holds for any stable system in steady state, not just M/M/1 queues. For example, a coffee shop averaging 15 customers inside with 120 arrivals per hour gives W = 15/120 = 0.125 hours, about 7.5 minutes.
How do you find the optimal number of servers in a queuing system?
Total cost per hour is waiting cost times queue length plus server cost times servers, C_w L_q + C_s s; evaluate it at one, two, three servers and stop where adding one more raises the total. In a bank with 18 arrivals per hour, a service rate of 20, $25 waiting cost and $15 per teller, one teller costs $217.50 an hour, two cost $35.71, and three cost $45.75, so two is optimal.

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