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- Identify or Define:

The assumptions of the two basic waiting-line

models.

- Explain or be able to use:

How to apply waiting-line models?

How to conduct an economic analysis of

queues?

Elements of Waiting Line Analysis

- Elements of a waiting line: arrivals, servers, waiting line.
- The Calling Population: Source of customers.
- Infinite: Assumes a large number of potential customers, e.g. grocery store, bank etc.
- Finite: Has a specific, countable number of potential customers, e.g. repair facility for trucks etc.
- The Arrival Rate: Poisson Distribution.
- Service Times: Negative Exponential Distribution.
- Arrival Rate is less than Service Rate: If not waiting line will continue to grow.

Elements of Waiting Line Analysis

- Queue Discipline & Length
- Discipline: Order in which customers are served. FCFS, LIFO, Random.
- Length:
- Infinite: Movie theater.
- Finite: bank teller driveway can accommodate only so many cars.
- Basic Waiting Line Structures:
- Single-Channel, Single-Phase: Post Office with 1-clerk.
- Single-Channel, Multiple-Phase: Post Office with more than 1-clerk.
- Multiple-Channel, Single-Phase: Nurse-Doctor (Checkup of patients).
- Multiple-Channel, Multiple-Phase: Several Nurse-Doctor teams.
- Operating Characteristics:
- L, Lq, W, Wq, P0, Pn etc.
- Describes the performance of the queuing system.
- Which the management uses to evaluate the system and make decisions.

Why is it Important?

- Customers regard waiting as a non-value added activity.
- Often associated with poor service quality, especially if there is a long wait.
- Even in organizations, having employees or work wait is a non-value added activity.
- Is a cost to the customers (Customer waiting cost).
- Costs the company either to minimize or eliminate waiting OR to lose customers.

- A waiting line system consists of two components:
- The customer population (people or objects to be processed).
- The process or service system.
- Whenever demand exceeds available capacity, a waiting line or queue forms.

- Finite versus Infinite populations:
- Is the number of potential new customers affected by the number of customers already in queue?
- Balking
- When an arriving customer chooses not to enter a queue because it’s already too long
- Reneging
- When a customer already in queue gives up and exits without being serviced
- Jockeying
- When a customer switches back and forth between alternate queues in an effort to reduce waiting time

Waiting lines occur naturally because of two reasons:

1. Customers arrive randomly, not at evenly placed times nor at predetermined times

2. Service requirements of the customers are variable. (Think of a bank for example)

Both Arrival & Service Times exhibit a high degree of variability.

Leads to

Temporarily Overloaded System

Waiting Line

System Idle

(No Customers and hence Waiting Line)

- Quantifiable costs:
- When the customers are internal (e.g employees waiting for making copies), salaries paid to the employees
- Cost of the space of waiting (e.g. patient waiting room)
- Loss of business (lost profits)
- Hard to quantify costs:
- Loss of customer goodwill
- Loss of social welfare (e.g. patients waiting for hospital beds)

- Waiting lines can be reduced by increasing capacity:
- More service counters or servers
- Adding workers to increase speed
- Install faster processing systems

COST!

Queuing Analysis

- Mathematical analysis of waiting lines
- Goal: Minimize Total Cost
- Service capacity costs
- Customer waiting costs (difficult to quantify)
- Work with an acceptable level of waiting & establish capacity to achieve that level of service.
- Traditional Approach: Balance the cost of providing a level of service capacity with the cost of customers waiting for service Minimizes Total Cost.
- Economic analysis can be done using either BEA or NPV.

Recap

- What is a waiting line?
- Why is there waiting?
- Elements of Waiting Line Analysis.
- Why is it important?
- Terminology.
- Cost of Waiting:
- Quantifiable.
- Non-Quantifiable.

Characteristics of Waiting Line

- The service system is defined by:
- The number of waiting lines
- The number of servers
- The arrangement of servers
- The arrival and service patterns
- The service priority rules

- Arrival rate:
- The average number of customers arriving per time period
- Modeled using the Poisson distribution
- Service time:
- The average number of customers that can be serviced during the same period of time
- Modeled using the exponential distribution

Modeling the Arrivals

Arrival

X

X

X

X

X

X

X

Inter-arrival Time

Unit Time Period

Inter-arrival Time = Time between two consecutive arrivals (a random quantity).

Number = Number of arrivals in a unit time period (also a random quantity).

How to represent this randomness??

Use Common Probability Distributions.

Modeling the Arrivals

Determine the arrival rate () = average number of arrivals per unit time period. Represent Number of arrivals with a Poisson probability distribution with rate ().

Represent Inter-arrival time with an Exponential probability distribution with mean

(1/ ).

Poisson Rate Exponential Inter-

of Arrivals Arrival Time

Modeling the Service

Determine the service rate () = number of customers that can be served per period.

Represent Service Time with an Exponential probability distribution with mean

(1/).

Poisson Service Rate Exponential Service Time

- Waiting lines systems can have single or multiple queues.
- Single queues avoid jockeying behavior & all customers are served on a first-come, first-served fashion (perceived fairness is high)
- Multiple queues are often used when arriving customers have differing characteristics (e.g.: paying with cash, less than 10 items, etc.) and can be readily segmented

- Single server or multiple/ parallel servers providing multiple channels
- Arrangement of servers (phases)
- Multiple phase systems require customers to visit more than one server
- Example of a multi-phase, multi-server system:

1

4

Arrivals

Depart

C

C

C

C

C

2

5

3

6

Phase 1

Phase 2

Four Types of Queuing Systems

Single Channel,

Single Phase

Single Channel,

Multiple Phase

Multiple Channel,

Single Phase

Multiple Channel,

Multiple Phase

- The average number of customers waiting in queue.
- The average number of customers in the system (multiphase systems).
- The average waiting time in queue.
- The average time in the system.
- The system utilization rate (% of time servers are busy).
- Probability that an arrival will have to wait.

Customers per unit time

Relationship between the two distributions

- An Exponential Service Time means that most of the time, the service requirements are of short duration, but there are occasional long ones. Exponential Distribution also means that the probability that a service will be completed in the next instant of time is not dependent on the time at which it entered the system.
- Poisson and Negative Exponential Distributions are alternate ways of presenting the same basic information.
- If service time is Exponentially distributed, the Service rate will have Poisson distribution.
- If customer arrival rate has a Poisson distribution, the inter-arrival time is Exponential.
- E.g.

If = 12 / hr --------> Average Service Time = 5 minutes.

If = 10 / hr --------> Average inter-arrival time = 6 minutes.

Poisson and Exponential Distributions can be derived from each-other.

For Poisson: Mean = Variance = .

For Exponential: Mean = 1 / & Variance = 1/ 2.

- The use of exponential distributions is a convention. (it makes mathematical analysis simpler)
- You have to justify these distributions used in the waiting line models. (by collecting data and then checking for a distribution that fits the data, via statistical tests such as goodness of fit test).
- Poisson arrivals is realistic but exponential service is not.

Basic (Steady State) Relationships

- Utilization: (should be <1)
- Average Number in Service:
- Average Number in Queue (Lq ): model dependent
- Average Number in System :
- Average Time Customers Wait in Queue :

Wait in System :

Basic (Steady State) Relationships

One of the most important relationship in Queuing Theory is called

Little’s Law.

Little’s Law:

Average Inventory (customers here)

= Flow rate (arrival rate here) x Cycle time (average waiting time in queue here)

Two Popular Waiting Line Models

1. Model 1: Single Channel

2. Model 3: Multiple Channel

- Single phase
- Poisson arrivals
- Exponential service times
- FCFS queue discipline
- No limit on the waiting line length

- A help desk in the computer lab serves students on a first-come, first served basis. On average, 15 students need help every hour. The help desk can serve an average of 20 students per hour.
- Based on this description, we know:
- Mu = 20 (exponential distribution)
- Lambda = 15 (Poisson distribution)

where M is number of servers

Waiting in Line

in the System

Waiting in the Line

Spends in the System

Model3

Lqformula is complicated. Use Table 19-4 on page 824-825 in Book OR page 324 in the Course Pack to read the Lq value.

Use the basic relationship formulas to calculate other service measures

Extending Model 1

Other measures from the basic formulas by replacing with eff= (1- PK) (Why?)

Limitations of Waiting Line Models

- Mathematical analysis becomes very complicated or intractable for more complex waiting lines. Some examples:
- Non-Poisson arrivals
- Non-exponential service times
- Complex customer behavior (e.g. customers switching between lines, or leaving after some time etc)
- multiple phase systems

Simulation can be useful analysis tool

Example

Customers arrive at a bakery at an average rate of 18 per hour on weekday mornings. The arrival distribution can be described by a Poisson distribution with a mean of 18. Each clerk can serve a customer in an average of four minutes; this time can be described by an exponential distribution with a mean of 4.0 minutes.

A) What are the arrival and service rates?

B) Compute the average number of customers being served at any time (assuming that system utilization is less than 1).

C) Suppose it has been determined that the average number of customers waiting in line is 3.6. Compute the average number of customers in the system (i.e. waiting in line or being served), the average time customers wait in line, and the average time in the system.

D) Determine the system utilization for M=2, 3 and 4 servers.

Example

An Airline is planning to open a satellite ticket desk in a new shopping plaza, staffed by one ticket agent. It is estimated that requests for tickets and information will average 15 per hour, and request rates will have a Poisson distribution. Service time is assumed to be exponentially distributed. Previous experience with similar satellite operations suggests that mean service time should average about three minutes per request. Determine each of the following:

A) System utilization.

B) Percent of time the server (agent) will be idle.

C) The expected number of customers waiting to be served.

D) The average time customers will spend in the system.

E) the probability of zero customers in the system and the probability of four customers in the system.

Example

Alpha Taxi and Hauling Company has seven cabs stationed at the airport. The company has determined that during the late-evening hours on weeknights, customers request cabs at a rate that follows the Poisson distribution with a mean of 6.6 per hour. Service time is exponential with a mean of 50 minutes per customer. Assume that there is one customer per cab and that taxis return to the airport. Find each of the following:

A) Average number of customers in line.

B) Probability of zero customers in the system.

C) Average waiting time for an arrival not immediately served.

D) Probability that an arrival will have to wait for service.

E) System Utilization.

Example

Alpha Taxi and Hauling Company also plans to have cabs at the new rail station. The expected arrival rate is 4.8 customers per hour, and the service rate (including return time to the rail station) is expected to be 1.5 per hour. How many cabs will be needed to achieve an average time in line of 20 minutes or less?

Example

Trucks arrive at a warehouse at a rate of 15 per hour during business hours. Crews can unload the trucks at a rate of five per hour. The high unloading rate is due to cargo being containerized. Recent changes in wage rates have caused the warehouse manager to reexamine the question of of how many crews to use. The new rates are: crew and dock cost is $100 per hour; truck and driver cost is $120 per hour.

Example

One of the features of a new machine shop will be a well-stocked crib. The manager of the shop must decide on the number of attendants needed to staff the crib. Attendants will receive $9 per hour in salary and fringe benefits. Mechanics’ time will be worth $30 per hour, which includes salary and fringe benefits plus two lost work time caused by waiting for parts. Based on previous experience, the manager estimates requests for parts will average 18 per hour with a service capacity of 20 requests per hour per attendant. How many attendants should be on duty if the manager is willing to assume that arrival and service rates will be Poisson-distributed? (Assume the number of mechanics is very large, so an infinite-source model is appropriate).

Example

The following is a list of service times for three different operations:

OperationsService Time

A 8 minutes

B 1.2 hours

C 2 days

A) Determine the service rate for each operation.

B) Would the calculated rates be different if there were interarrival times rather than service times?

Example

A small town with one hospital has two ambulances to supply ambulance service. Requests for ambulances during non-holiday weekends average 0.8 per hour and tend to be Poisson distributed. Travel and assistance time averages one hour per call and follows an exponential distribution.

Find:

A. System utilization.

B. The average number of customer waiting.

C. The average time customers wait for an ambulance.

D. The probability that both ambulances will be busy when a call comes in.

Example

Trucks are required to pass through a weighing station so that they can be checked for weight violations. Trucks arrive at the station at the rate of 40 an hour between 7 p.m. and 9 p.m. Currently two inspectors are on duty during those hours, each of whom can inspect 25 trucks an hour:

A. How many trucks would you expect to see at the weighing station, including those being inspected?

B. If a truck was just arriving at the station, about how many minutes could the driver expect to be at the station?

C. What is the probability that both inspectors would be busy at the same time?

D. How many minutes, on average, would a truck that is not immediately inspected have to wait?

Example

The manager of a regional warehouse must decide on the number of loading docks to request for a new facility in order to minimize the sum of dock costs 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 loading crew represents a cost of $1,100 per day.

A. How many docks should be requested if trucks arrive at the rate of four per day, each dock can handle five trucks per day, and both rates are Poisson?

B. An employee has proposed adding new equipment that would speed up the loading 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?

Example

The parts department of a large automobile dealership has a counter used exclusively for mechanics’ requests for parts. The time between requests can be modeled by a negative exponential distribution that has a mean of five minutes. A clerk can handle requests at a rate of 15 per hour, and this can be modeled by a Poisson distribution that has a mean of 15. Suppose there are two clerks at the counter.

A. On average, how many mechanics would be at the counter, including those being served?

B. What is the probability that a mechanic would have to wait for service?

C. If a mechanic has to wait, how long would the average wait be?

D. What percentage of time are the clerks idle?

E. If clerks represent a cost of $20 per hour and mechanics a cost of $30 per hour, what number of clerks would be optimal in terms of minimizing total cost?

Example

Students arrive at the Administrative Services Office at an average of one every 15 minutes, and their requests take an average of 10 minutes to be processed. The service counter is staffed by one clerk, Judy Gumshoes, who works eight hours per day. Assume Poisson arrivals and exponential service times.

A. What percentage of time is Judy idle?

B. How much time, on average, does a student spend waiting in line?

C. How long is the (waiting) line on average?

D. What is the probability that an arriving student (just before entering the Administrative Services Office) will find at least one other student waiting in line?

Example

The managers of the Administrative Services Office estimate that the time a student spends waiting in line costs them (due to goodwill loss and so on) $10 per hour. To reduce the time a student spends waiting, they know that they need to improve Judy’s processing time (previous example).

They are currently considering the following two options:

A. Install a computer system, with which Judy expects to be able to complete a student request 40 percent faster (from 2 minutes per request to 1 minute and 12 second, for example).

B. Hire another temporary clerk, who will work at the same rate as Judy?

If the computer costs $99.50 to operate per day, while the company clerk gets paid $75 per day, is Judy right to prefer the hired help? Assume Poisson arrival and Exponential service times.

Example

The manager of a grocery store in the retirement community of Sunnyville is interested in providing good service to the senior citizens who shop at his store. Presently, the store has a separate checkout counter for senior citizens. On average, 30 senior citizens per hour arrive at the counter, according to a Poisson distribution, and are served at an average rate of 35 customers per hour, with exponential service times.

Find the following operating characteristics:

A. Probability of zero customers in the system.

B. Utilization of the checkout clerk.

C. Number of customers in the system.

D. Number of customers in line.

E. Time spent in the system.

F. Waiting time in line.

Example

The manager of the Sunnyville grocery in the above example wants answers to the following questions:

A. What service rate would be required to have customers average only eight minutes in the system?

B. For that service rate, what is the probability of having more than four customers in the system?

C. What service rate would be required to have only a 10 percent chance of exceeding four customers in the system?

Example

The management of the American Parcel Service terminal in Verona, Wisconsin, is concerned about the amount of time the company’s trucks are idle, waiting to be loaded. The terminal operates with four unloading bays. Each bay requires a crew of two employees, and each crew costs $30 per hour. The estimated cost of an idle truck is $50 per hour. Trucks arrive at an average rate of three per hour, according to a Poisson distribution. On average a crew can unload a semitrailer rig in one hour, with exponential service times.

A. Probability of zero customers in the system.

B. Average utilization of the server.

C. Average number of customers in line & in the system.

D. Average waiting time in line & the system.

E. What is the total hourly cost of operating the system?

Example

The Mega Multiplex Theater has three concession clerks serving customers on a FCFS basis. The service time per customer is exponentially distributed with an average of 2 minutes per customer. Concession customers wait in a single line in a large lobby, and arrivals are Poisson distributed with an average of 81 customers per hour. Previews run for 10 minutes before the start of each show. If the average time in the concession area exceeds 10 minutes, customers become dissatisfied.

A. What is the average utilization of the concession clerk?

B. What is the average time spent in the concession area?

Example

A popular attraction at the Montreal Old Port is a street artist who will paint a caricature in about 5 minutes, exponentially distributed. Customers are willing to wait, but when there are more than 10 waiting, they are turned away. The arrival rate is 20 per hour.

A. What percent of customers are turned away?

B. What is the rate at which customers are turned away?

C. What is the average number of customers in line & in the system?

D. What is the average waiting time in line and in the system?

Example

Slick’s Quick Lube is a one-bay service facility next to a busy highway. The facility has space for only one vehicle in service and three vehicles lined up to wait for service. There is no space for cars to line up on the busy adjacent highway, so if the waiting line is full (3 cars), prospective customers must drive on.

The mean time between arrivals for customers seeking lube service is 3 minutes. The mean time required to perform to lube operation is 2 minutes. Both the inter-arrival times and the service times are exponentially distributed. The maximum number of vehicles in the system is four. Determine the average waiting time, the average queue length, and the probability that a customer will have to drive on.

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