Welcome to Comm 305!

Commerce 341 Operations ManagementQueuing Theory

Fall 2015Geoff PondIntroductionSingle Channel, Single Server w/ Exp. Service TimeSingle Channel, Single Server w/ Constant Service TimeSingle Channel, Multi-Server w/ Exp. Service TimePsychology of QueuesStrategies

AgendaLike it or not, it happens everyday:

Vehicles arriving at a traffic lightCustomers arriving at Tim HortonsContainer ships arriving at a port and waiting for a tug boatLineups at customs/immigration/passport controlAircraft waiting on a taxiway for runway clearanceStudents waiting in line outside Stages on a Friday night at 11pmStudents waiting at Smokies for some 2am poutinePatients arriving at KGH ER to have their stomach pumped at 3am

IntroductionWhy is it important to study?

Customers waiting for extended periods may balkThere may be insufficient space to accommodate the queue length (causing safety problems) Efficiency/capacity implications (bottlenecks)Effectiveness implications (e.g., patient conditions may deteriorate)

Introduction

Just imagine the monetary value of the fuel being burned as these aircraft wait for take-off (Atlanta International)Introduction

Our goal is to find the sweet spot that minimizes total cost.Different Queue Models

A / S / k

where:Arepresents the distribution of arrival timesS represents the distribution of service timeskrepresents the number of servers

A and B are normally either: M meaning Markov or memoryless Dmeaning deterministic or fixed service timeNaming ConventionMost models deal with Markovian or memoryless processes.

Arrival rates are typically described using the Poisson distribution.

Service times are typically described using the Exponential distribution.Naming Conventions

M / M / 1Naming ConventionsOne server (e.g., a single cashier)Markovian service, i.e., exponentially distributed service timesRandom arrivals, i.e., Poisson arrival rate

M / M / 3

Infinite SourceThere is a large calling population

Finite SourceThe calling population is small enough that as the number of units in the system increases, a change is observed in the arrival rate of new unitsIn the extreme case, if a calling population where 5 units and all 5 units were in the system, the arrival rate would become 0 But obviously it was initially higher than that (or the five units wouldnt have arrived in the first place)Queuing SourceTypical Waitline Parameters

Note that problems dont necessarily give you the data in a rate in these cases it will be up to you to make the conversion.M / M / 1ExampleConsider a bank where customers arrive at a rate of 20 customers per hour. On average, a customer is served every 2 minutes.

M / M / 1

ExampleConsider a bank where customers arrive at a rate of 20 customers per hour. On average, a customer is served every 2 minutes. M / M / 1

ExampleConsider a bank where customers arrive at a rate of 20 customers per hour. On average, a customer is served every 2 minutes. M / M / 1

ExampleConsider a bank where customers arrive at a rate of 20 customers per hour. On average, a customer is served every 2 minutes. M / M / 1

ExampleConsider a bank where customers arrive at a rate of 20 customers per hour. On average, a customer is served every 2 minutes. M / M / 1ExampleConsider a bank where customers arrive at a rate of 20 customers per hour. On average, a customer is served every 2 minutes.

M / M / 1ExampleConsider a bank where customers arrive at a rate of 20 customers per hour. On average, a customer is served every 2 minutes.

WARNINGThe equations we just discussed are specific to M / M / 1 queuing systems. They are NOT general equations, meaning that they cannot necessarily be applied to M / D / 1 or M / M / >1 systems that we will look at next. M / D / 1ExampleA vending machine dispenses hot chocolate or coffee. Service duration is 30 seconds per cup and is constant. Customers arrive at a mean rate of 80 per hour (following a Poisson distribution). Also assume that each customer buys only one cup. What is the average number of customers waiting in line?This is the only equation that is different from M / M / 1M / D / 1ExampleA vending machine dispenses hot chocolate or coffee. Service duration is 30 seconds per cup and is constant. Customers arrive at a mean rate of 80 per hour (following a Poisson distribution). Also assume that each customer buys only one cup. How long do customers spend in line?M / D / 1ExampleA vending machine dispenses hot chocolate or coffee. Service duration is 30 seconds per cup and is constant. Customers arrive at a mean rate of 80 per hour (following a Poisson distribution). Also assume that each customer buys only one cup. What is the server utilization?

Regardless of the queuing model, the following equations are general. Theyre also easy so they can be quite handy, especially when dealing with otherwise complicated problems.Littles Flow EquationsMany of a banks customers use its automated teller machine (ATM). During the early evening hours in the summer months, customers arrive at the ATM at the rate of one every other minute s(assume Poisson). Each customer spends an average of 90 seconds completing his or her transaction. Transaction times are exponentially distributed. Determine:

The average time customers spend at the machine, including waiting in line and completing transactions.The probability that a customer will not have to wait upon arrival the ATM.Utilization of the ATM.The probability of three customers waiting in line.

You Try One!M / M / >1

M / M / >1ExampleA small town with one hospital has two ambulances. Requests for an ambulance during weekday mornings average .8 per hour and tend to be Poisson. Travel and loading/unloading time averages one hour per call and follows an Exponential Distribution. Find the server utilization.M / M / >1ExampleA small town with one hospital has two ambulances. Requests for an ambulance during weekday mornings average .8 per hour and tend to be Poisson. Travel and loading/unloading time averages one hour per call and follows an Exponential Distribution. Find the average number of patients waiting.M / M / >1ExampleA small town with one hospital has two ambulances. Requests for an ambulance during weekday mornings average .8 per hour and tend to be Poisson. Travel and loading/unloading time averages one hour per call and follows an Exponential Distribution. Find the average number of patients waiting.M / M / >1ExampleA small town with one hospital has two ambulances. Requests for an ambulance during weekday mornings average .8 per hour and tend to be Poisson. Travel and loading/unloading time averages one hour per call and follows an Exponential Distribution. Find the average time patients wait for an ambulance.The manager of a new regional warehouse of a company must decide on the number of loading docks to request in order to minimize the sum of dock-crew and driver-truck costs. The manager has learned that each driver-truck combination represents a cost of $300 per day and that each dock plus crew represents a cost of $1,100 per day. How many docks should she request if trucks will arrive at an average rate of four per day, each dock can handle an average of five trucks per day, and both rates are Poisson?Now You Try One!The manager of a new regional warehouse of a company must decide on the number of loading docks to request in order to minimize the sum of dock-crew and driver-truck costs. The manager has learned that each driver-truck combination represents a cost of $300 per day and that each dock plus crew represents a cost of $1,100 per day. An employee has proposed adding new equipment that would speed up the service rate to 5.71 trucks per day. The equipment would cost $100 per day for each dock. Should the manager invest in the new equipment?Now You Try One!As you might guess, no one is sitting in corporate backrooms cranking out these equations. Instead, sim is often usedIn Practice

ArenaBasic licensing: $2,495 / seatStudent license: FREE!!https://www.arenasimulation.com/academic/students ExtendSimBasic licensing:Unoccupied Time Feels Longer than Occupied TimePre-Process Waits Feel Longer than In-Process WaitsAnxiety Makes Waits Seem LongerUncertain Waits Are Longer than Known, Finite WaitsUnexplained Waits Are Longer than Explained WaitsUnfair Waits are Longer than Equitable Waits

Maister, D.H. (1984). Psychology of Waiting Lines, Harvard Business Review.

Psychology of QueuesKeep progress moving. Were often satisfied as long as were getting closer to the goal, even if its slowly. Avoid designing queues where the consumers move away from the service desk. If necessary, do it as late as possible in the queueGet consumers in the system ASAP. Consider a tiered (or multi-phase) serviceSocial Justice: FIFO is viewed as being most equitable but dont always happen (think about multiple queues each associated with an independent server where one moves faster than the other)

Soman, D. (2013). The Waiting Game: The Psychology of Time and its Effects on Service Design, Rotman Magazine. Psychology of QueuesKnow consumer expectations! How long will consumers accept waiting in line?Keep service rates consistent when possible.Distract them (music, televisions, magazines, etc.)Keep customers informedSelf-serviceStaff not directly involved in serving customers should be kept out-of-sightStay friendly!

StrategiesReview the supplemental chapter on Queuing Theory (7S) from McGraw Hill Connect. Let me know if you have questions!!!Try the following problems from the end of the supplemental chapter:

Problem 1Problem 7Problem 17Problem 18

Read Chapter 10 in preparation for next class.

Before next class.