...Waiting Lines & QueuingModels
American Military University
Business 312
For my project on other operations research techniques I have decided to research waiting lines and queuingmodels. My interest in this application stems from my personal dislike for standing in lines and waiting on hold while on the phone. This is virtually my only pet peeve; nothing aggravates me faster than standing in a line or waiting on hold. Like most people I go out of my way to avoid lines, using strategies such as arriving early or visiting during non-peak times. However, before investigating this topic, I had no idea there was a specific science behind the madness.
Queuingmodels are important applications for predicting congestion in a system. This can encompass everything from a waiting line at pharmacy to traffic flow at a busy intersection. This is important because it can impact businesses in unforeseen ways. Customers may begin to believe that they are wasting their time when they are forced to wait in line for service and continued delays may begin to negatively influence their shopping preferences.
Organizations design their waiting line systems by weighing the consequences of having a customer wait in line, versus the costs of providing more service capacity. Queuing theory provides a variety of analytical models that can be used to help decision...

...server Because in particular for a service-operation, ִwe don’t have the luxury of satisfying demand from inventory
3
Tradeoff…
Service Capacity Cost ִhiring servers ִtraining servers Waiting Cost ִcustomer dissatisfaction ִloss of potential customers ִcapital tied up in WIP ִstorage facility cost
Balance …
4
5
Description of queues
server
Customer arrivals
queue
Customer departures
system
Priority rules: e.g., first-come-first-serve
6
Types ofQueuing Systems
Single stage system
multiple stage system
parallel single stage system
multi-channel single stage system
7
Other Arrival Characteristics
Size of Arrival Units Degree of patience Balking Reneging Jockeying
Managerial Issues of Queuing Systems
System Design • how many servers • arrangement of queue • fast vs. slow server • size of waiting area
System Management/Operations • management of arrivals • customer perception of waiting times
9
Managerial Issues of Queuing Systems (cont’d) Performance Evaluations • Average number of customers in the queue: Nq • Average waiting time (time in the queue): Tq • Average number of customers in the system: N or Ns • Average time in the system: T or Ts • System capacity utilization: ρ • Probability of 0 customers in the system: P0 • Probability of k customers in the system: Pk • Probability of waiting time less than a specific amount
10
Performance Measures:...

...Problem: B&K groceries operates with three checkout counters. The manager uses the
following schedule to determine the number of counters in operation, depending on the number
of customers in line:
Number of customers in store Number of counters in operation
1 to 3 1
4 to 6 2
More than 6 3
Customers arrive in the counters area according to a Poisson distribution with a mean rate
of 10 customers per hour. The average checkout time per customer is exponential with mean
12 minutes. Determine the steady-state probability pn of n customers in the checkout area.
Solution:
λn = λ = 10 customers per hour, n = 0, 1, 2, . . .
µn =
60
12 = 5 customers per hour, n = 1, 2, 3
2 × 5 = 10 customers per hour, n = 4, 5, 6
3 × 5 = 15 customers per hour, n = 7, 8, . . .
Then:
p1 =
10
5
p0 = 2p0
p2 =
10 × 10
5 × 5
p0 = 4p0
p3 =
10 × 10 × 10
5 × 5 × 5
p0 = 8p0
p4 =
10 × 10 × 10 × 10
5 × 5 × 5 × 10
p0 = 8p0
p5 =
10 × 10 × 10 × 10 × 10
5 × 5 × 5 × 10 × 10
p0 = 8p0
p6 =
10 × 10 × 10 × 10 × 10 × 10
5 × 5 × 5 × 10 × 10 × 10
p0 = 8p0
pn≥7 =
10 × 10 × 10 × 10 × 10 × 10
5 × 5 × 5 × 10 × 10 × 10 10
15n−6
p0 = 8
2
3
n−6
p0
1
Example 1:
The weather in Amman can be cloudy (C), sunny (S), or rainy (R). Records over the past 16
days are
{CCRRSSCCCRCSSRCR}
Based on these records, use a Markov chain to determine the probability that a typical day in
Amman will be cloudy, sunny, or rainy?...

...Simulation Technique in QueuingModel for ATM
Facility
Journal Name: International Journal of Applied Engineering Research, Dindigul
Volume 1, No 3
Date of publications: 2010
Pages of article: Pages 469-482 (14 pages)
JOURNAL SUMMARY
1.0 Issues/ Problem Statement:
Most of the ATMs have the problem of long queue of customers to undergo simple transaction at the peak hours and remain idle due to the lack of customer entry at the off peak hours.
2.0 Objectives:
1. To develop a simulation model to reduce the waiting time of customers and the total operation cost related to ATM installation.
2. To determine whether only one machine is required to fulfill the need or two more machines are needed to be installed to give comfort to customers who are really of short period of time.
3. To develop an efficient procedure for ATM queuing problem
3.0 Literature Review:
Apart from ATM problem, simulation with queuingmodel had been used for various applications too:
According to Pieter Tjerk de Boer (1983), substantial focus has been dedicated to the estimation of overflow probabilities in queuing networks. A different adaptive method has applied to queuing problems than in the present work with few simple models been considered.The...

...REVISED
M14_REND6289_10_IM_C14.QXD 5/12/08 1:01 PM Page 218
218
CHAPTER 14
WAITING LINE
AND
QUEUING THEORY MODELS
Alternative Example 14.3: A new shopping mall is considering setting up an information desk manned by two employees. Based on information obtained from similar information desks, it is believed that people will arrive at the desk at the rate of 20 per hour. It takes an average of 2 minutes to answer a question. It is assumed that arrivals are Poisson and answer times are exponentially distributed. a. Find the proportion of the time that the employees are idle. b. Find the average number of people waiting in the system. c. Find the expected time a person spends waiting in the system. ANSWER: (servers). a. P 20/hour, 30/hour, M 2 open channels
SOLUTIONS TO DISCUSSION QUESTIONS AND PROBLEMS
14-1. The waiting line problem concerns the question of ﬁnding the ideal level of service that an organization should provide. The three components of a queuing system are arrivals, waiting line, and service facility. 14-2. The seven underlying assumptions are: 1. Arrivals are FIFO. 2. There is no balking or reneging. 3. Arrivals are independent. 4. Arrivals are Poisson. 5. Service times are independent. 6. Service times are negative exponential. 7. Average service rate exceeds average arrival rate. 14-3. The seven operating characteristics are: 1. Average number of customers in the system (L) 2. Average time...

...Study for Restaurant QueuingModel
Mathias Dharmawirya
School of Information Systems Binus International – Binus University Jakarta, Indonesia mdharmawirya@binus.edu
Erwin Adi
School of Computer Science Binus International – Binus University Jakarta, Indonesia eadi@binus.edu busy fast food restaurant [3], as well as to increase throughput and efficiency [5]. This paper uses queuing theory to study the waiting lines in Sushi Tei Restaurant at Senayan City, Jakarta. The restaurant provides 20 tables of 6 people. There are 8 to 9 waiters or waitresses working at any one time. On a daily basis, it serves over 400 customers during weekdays, and over 1000 customers during weekends. This paper seeks to illustrate the usefulness of applying queuing theory in a realcase situation. II. QUEUING THEORY In 1908, Copenhagen Telephone Company requested Agner K. Erlang to work on the holding times in a telephone switch. He identified that the number of telephone conversations and telephone holding time fit into Poisson distribution and exponentially distributed. This was the beginning of the study of queuing theory. In this section, we will discuss two common concepts in queuing theory. A. Little’s Theorem Little’s theorem [7] describes the relationship between throughput rate (i.e. arrival and service rate), cycle time and work in process (i.e. number of customers/jobs in the system)....

...Introduction
Being in a queue (waiting line) is an inevitable fact of our daily life, such as waiting for checkout at a supermarket, or waiting to make a bank deposit. Queuing theory, started with research by Agner Krarup Erlang, is used to examine the impact of management decisions on these waiting lines (Anderson et.al, 2009). A basic QueuingModel structure consists of three main characteristics, namely behaviour of arrivals, queue discipline, and service mechanism (Hillier and Lieberman, 2001).
In this assignment, New England Foundry’s queuing problem will be solved in Excel, and then, time and cost savings will be identified.
First of all, current and new situation will be analysed in order to demonstrate the queuingmodel by using Kendall’s Notation (for the current queuing problem, queuingmodel is M/M/s). After that, arrival rate, queue size, and service rate will be defined, and added-in Excel file (Queuing models.xlsx). The results will be discussed at the end.
Description
New England Foundry (NEF) produces four different types of woodstoves for home use and additional products that are used with these four stoves.
Due to the increase in energy prices, George Mathison president of the company wants to change the layout to increase the production of their bestselling type of Warmglo III.
NEF has several operations...

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