Free Essay

Queuing Theory

In: Business and Management

Submitted By tzaditz
Words 2526
Pages 11
Queuing Theory

Queuing Theory

• Queuing theory is the mathematics of waiting lines. • It is extremely useful in predicting and evaluating system performance. • Queuing theory has been used for operations research. Traditional queuing theory problems refer to customers visiting a store, analogous to requests arriving at a device.

Long Term Averages
• Queuing theory provides long term average values. • It does not predict when the next event will occur. • Input data should be measured over an extended period of time. • We assume arrival times and service times are random. • • • •

Assumptions
Independent arrivals Exponential distributions Customers do not leave or change queues. Large queues do not discourage customers. Many assumptions are not always true, but queuing theory gives good results anyway

Queuing Model
Q W λ Tw
Tq
S

Interesting Values
• Arrival rate (λ) — the average rate at which customers arrive. • Service time (s) — the average time required to service one customer. • Number waiting (W) — the average number of customers waiting. • Number in the system (Q) — the average total number of customers in the system.

More Interesting Values
• Time in the system (Tq) the average time each customer is in the system, both waiting and being serviced. Time waiting (Tw) the average time each customer waits in the queue. Tq = Tw + s

Arrival Rate
• The arrival rate, λ, is the average rate new customers arrive measured in arrivals per time period. Common units are access/second • The inter-arrival time, a, is the average time between customer arrivals. It is measured in time per customer. A common unit would be seconds/access.



a=1/λ

Random Values
• We assume that most of the events we are interested in occur randomly.
– Time of a request to a device – Time to service a request – Time user makes a request

Exponential Distribution
• Many of the random values are exponentially distributed. Frequency of Occurrence = e-t • There are many small values and a few large values. • The inter-arrival time of customers is naturally exponentially distributed.

• Although events are random, we may know the average value of the times and their distribution. • If you flip a coin, you will get heads 50% of the time.

Exponential Distribution
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Poisson Arrival Rate
If customers are arriving at the exponentially distributed rate λ, then the probability that there will be k customers after time t is:

(λt ) k −λt Pk (t ) = e k!

Math Notes

Poisson Example
• A networked printer usually gets 15 print jobs every hour. The printer has to be turned off for 10 minutes for maintenance. What is the probability that nobody will want to use the printer during that time?

0! = 1! = 1 X0 = 1 X1 = X

Poisson Solution
• A networked printer usually gets 15 print jobs every hour. The printer has to be turned off for 10 minutes for maintenance. What is the probability that nobody will want to use the printer during that time? • The arrival rate is 15/60 = 0.25 jobs/min.

Expected Number of Arrivals
If customers are arriving at the exponentially distributed rate λ, how many customers should you expect to arrive in time t?

Expected = λ * t
For the printer problem with an arrival rate λ = 0.25, in 10 minutes we should expect 2.5 jobs to arrive

P0 (10) =

(0.25 * 10) 0 −0.25*10 e = 0.082 0!

Queuing Models
Queuing systems are usually described by three values separated by slashes
Arrival distribution / service distribution / # of servers

Common Models
• The simplest queuing model is M/M/1 where both the arrival time and service time are exponentially distributed. • The M/D/1 model has exponentially distributed arrival times but fixed service time. • The M/M/n model has multiple servers.

where: • M = Markovian or exponentially distributed • D = Deterministic or constant. • G = General or binomial distribution

Why is there Queuing?
• The arrivals come at random times. • Sometimes arrivals are far apart. Sometimes many customers arrive at almost the same time. When more customers arrive in a short period of time than can be serviced, queues form. • If the arrival rate was not random, queues would not be created.

Utilization
• Utilization (represented by the Greek letter rho, ρ) is the fraction of time the server is busy. • Utilization is always between zero and one

0≤ρ≤1
• If a bank teller spends 6 hours out of an 8 hour day counting money, her utilization is 6/8 = 0.75

Calculating Utilization
• Utilization can be calculated from the arrival rate and the service time.

Little’s Formula
• The number in the system is equal to the arrival rate times the average time a customer spends in the system.

ρ = λ*s

Q = λ * Tq
• This is also true for just the queue.

It is important that the units of both the arrival rate and the service time be identical. It may be necessary to convert these values to common units.

W = λ * Tw

M/M/1 Formulas

Application of Little’s Formula
• Multiplying the formulas on the left by λ gives the formula on the right.

s Tq = 1− ρ
Tw = sρ 1− ρ

Q=

ρ
1− ρ

W=

ρ

2

λTq =

1− ρ

λs ρ = =Q 1− ρ 1− ρ

20 18 16 14 12 Tq 10 8 6 4 2 0 0 0.1 0.2 0.3 0.4 0.5 Utilization 0.6 0.7 0.8 0.9 1

Solution Process
1. Determine what quantities you need to know. 2. Identify the server 3. Identify the queued items 4. Identify the queuing model 5. Determine the service time 6. Determine the arrival rate 7. Calculate ρ 8. Calculate the desired values

Example
• Consider a disk drive that can complete an average request in 10 ms. The time to complete a request is exponentially distributed. Over a period of 30 minutes, 117,000 requests were made to the disk. How long did it take to complete the average request? What is the average number of queued requests?

Solution
• Determine what quantities you need to know.
– The average request time is Tq – The number of queued jobs is W

• Identify the server
– The disk drive is the server

• Identify the queued items
– Disk requests

• Identify the queuing model
– M/M/1

Solution (cont.)
• Determine the service time
S = 10 ms = 0.01 sec / request

Solution (cont.)
• Time to complete the average request

• Determine the arrival rate λ = 117,000 request / (30 min * 60 sec/min) = 65 requests / sec

TQ =

s 0.01 = = 28.6ms 1 − ρ 1 − 0.65
0.65 2 ρ2 = = 1.21 1 − ρ 1 − 0.65

• Calculate ρ ρ = λ*s = 0.01 sec/request * 65 req/sec = 0.65

The average length of the queue

W=

Number in the System
• The value Q represents the average number of jobs in the system, both waiting and being served. • There are not always Q jobs in the system. Sometimes there are more, sometimes less. Q is the average.

Queue Size Probabilities
The probability that there are exactly N jobs in the system is given by

Prob[Q = N ] = (1 − ρ ) ρ N
Summing the probabilities for individual cases gives the probability of N or less customers in the system

prob[Q ≤ N ] = ∑ (1 − ρ ) ρ i i =0

N

Large Queue Probabilities
• The probability that there are more than N customers in the system is the sum of the probabilities from N-1 to ∞. • Remembering that the sum of all probabilities is one, the probability that there are more than N customers in the system is:

Example Continued
• In the previous example, what is the probability that a request does not get queued? • A job can get serviced immediately if there are only zero or one jobs in the system.
P[Q = 0or1] = (1 − ρ ) * ρ 0 + (1 − ρ ) * ρ 1 = 0.35 + .2275 = 0.58

prob[Q > N ] = 1 − ∑ (1 − ρ ) ρ i i =0

N

Accuracy and Significant Digits
• Just because my calculator displays a 10 digit number does not mean the answer is accurate to 10 digits. • Your answer can only be as accurate as your input data. If your data has three significant digits, your answer cannot have more than three digits. • Always use as much accuracy as possible in these calculations and round off only at the end.

Constant Service Time
• In some systems the service time is always a constant. The M/D/1 model is used for constant service time. • There is less randomness in the system. • The wait time will be less.

M/D/1 Formulas
Tq = s(2 − ρ ) 2(1 − ρ )
Q=

M/D/1 Example

• An ATM network sends 53 byte packets over a 155 Mb/sec line. It always takes 2.74 ms to send a packet. Each second 145,000 packets are sent. How long does a packet wait to be sent?

ρ2
2(1 − ρ )

Tw =

sρ 2(1 − ρ )

W=

ρ2
2(1 − ρ )

M/D/1 Solution
• Determine what quantities you need to know. The average time spent in the queue, Tw. • Identify the server The transmission line. • Identify the queued items Packets (not bits or bytes) • Identify the queuing model M/D/1

M/D/1 Solution
• Determine the service time 2.74x10-6 seconds • Determine the arrival rate 145,000 packets/second • Calculate ρ ρ = 145,000 * 2.74x10-6 = 0.3973 • Calculate the desired values
Tw = sρ 2.75 *10 −6 * 0.3973 = = 9.03 *10 −7 sec 2(1 − ρ ) 2(1 − 0.3973)

Multiple Servers
S

Multiple Servers
• Customers arrive and join a single queue. • Whenever any of the servers is idle, it serves the first customer on the single queue. • All of the servers must be identical. Any customer can be served by any server. • When there are N servers, the model is

λ

S

S

M/M/N

Multiple Server Utilization
• The server utilization for an N server system is:

Intermediate Value K
• To make calculations easier, we first compute the value K.

ρ = λs / N

K=

This is the average utilization for all N servers.

∑ ∑

N −1 ( λs ) i

i! i =0 N ( λs ) i i = 0 i!

K Calculation
• The first term (i = 0) is always 1 • Note that the value in the denominator is equal to the numerator plus the last term. • Since the denominator is always larger than the numerator, the value K must always be less than 1. • The value K is an intermediate that simplifies calculations. It has no intrinsic meaning.

Multiple Servers Busy
The probability that all servers are busy is

C=

1− K λsK 1− N

This is the probability that a new customer will have to wait in the queue.

M/M/N formulas
Tq = Cs +s N (1 − ρ )
Q=C

Example ρ + λs
• Assume that you have a printer that can print an average file in two minutes. Every two and a half minutes a user sends another file to the printer. How long does it take before a user can get their output?

1− ρ

Tw =

Cs N (1 − ρ ) note: W =C ρ = λs / N

ρ
1− ρ

Slow Printer Solution
• Determine what quantities you need to know. How long for job to exit the system, Tq • Identify the server The printer • Identify the queued items Print job • Identify the queuing model M/M/1

Slow Printer Solution
• Determine the service time Print a file in 2 minutes, s = 2 min • Determine the arrival rate new file every 2.5 minutes. λ = 1/ 2.5 = 0.4 • Calculate ρ ρ = λ * s = 0.4 * 2 = 0.8 • Calculate the desired values Tq = s / (1- ρ) = 2 / (1 - 0.8) = 10 min

Add a Second Printer
• To speed things up you can buy another printer that is exactly the same as the one you have. How long will it take for a user to get their files printed if you had two identical printers? • All values are the same, except the model is M/M/2 and ρ = λ * s / 2 = 0.4

Calculate K
K=

∑ ∑

N −1 ( λs ) i i! i =0 ( λs ) i i = 0 i! N

(λ s ) 0 (λ s ) 1 + 0! 1! = 0 1 (λs ) (λ s ) (λ s ) 2 + + 0! 1! 2!

K = 0.849057

Calculate M/M/N Solution
1− K C= = 0.22857 λsK 1− N

Faster Printer
• Another solution is to replace the existing printer with one that can print a file in an average of one minute. How long does it take for a user to get their output with the faster printer? • M/M/1 queue with λ = 0.4 and s = 1.0 Tq = s / (1- ρ) = 1 / (1 - 0.4) = 1.67 min A single fast printer is better, particularly at low utilization. 6X better than slow printer.

Tq =

Cs + s = 2.57 min N (1 − ρ )

Note that with twice as many printers this example runs about 4X as fast.

Multiple Arrival Streams
• Exponentially distributed arrival streams can be merged. The total arrival rate is the sum of the individual arrival rates.

Dividing Customer Streams
• An exponentially distributed arrival stream can be divided. The sum of the separated arrival rates must equal to original arrival rate.

λa λa + λb

λ/2 λ λ/2

λb

Linking Multiple Queues
• The exit rate of a system is equal to the arrival rate. • The output from one queuing system can feed into another.

Multiple Queue Example
• Consider a network with a computer connected to a router which is connected to a server. The computer can send a packet in 12 ms while the router can send it to the server in 7 ms. Programs on the computer generate 40 packets/second. The router receives a total of 100 packets/second. • How long does it take for a packet to get to the server?
40 pkt/sec 100 pkt/sec 12 ms Router 7 ms Server Computer

λ

λ

The time through the system is the sum of the time through each queuing component.

Multiple Queue Solution
• Determine what quantities you need to know Sum of Tq for both networks • Identify the servers The computer transmitter and the router • Identify the queued items Packets • Identify the queuing model both M/M/1 queues

Multiple Queue Solution
• Determine the service time 12ms for the computer, 7 ms for the router • Determine the arrival rate 40 pkt/sec for computer, 100 pkt/sec for router • Calculate ρ ρ = 40*0.012 = 0.48 ρ =100*0.007 = 0.7 • Calculate the desired values =0.012/(1-0.48) = 0.0231 sec Tq Tq =0.007/(1-0.7) = 0.0233 sec Total=46.4ms computer router computer router

Reusing a Server
• Consider a file server. Requests use the network to get to the server, then use the disk, then the network again to get back. • The load on the network is doubled.

λ λ

Net Disk

λ

λ

Similar Documents

Free Essay

Queuing Theory

...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 Queuing Model 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 queuing model by using Kendall’s Notation (for the current queuing problem, queuing model 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 in order to produce woodenstoves which are illustrated as a flow diagram in Figure 1. Current State Analysis Current layout offers one counter...

Words: 1225 - Pages: 5

Premium Essay

Queuing Theory

...Table of contents 1. Introduction 2. Arrival Pattern of customers 3. Service Patterns 4. System Capacity 5. Number of Service Channels 6. Queue Discipline 7. Queuing Cost 8. The Four Models 9. Model-1(Single Channel Queuing Model) 10. Model-2 (Multiple-Channel Queuing) 11. Model-3 (Constant-Service-Time) 12. Model-3 (Constant-Service-Time) 13. Simulation 14. Conclusion Abstract This report is about queuing theory, it’s application and analysis. Queuing theory has a vast number of applications starting from the simplest day to day life examples to complicated computer algorithms. To further explain the queuing theory analysis we have used simulation of an example from our case study. We have done an in depth analysis of the four queuing theory models and chosen one of them for the simulation. The results can be helpful in improving the overall performance of the manufacturing facility. Introduction According to U. Narayan Bhat waiting line are a phenomena through which businesses and facilities can be helped in an orderly manner. There are several ways to forma queue (waiting line), for instance when people wait to get a boarding pass from an airline counter, there can be 3 service stations (airline counters) and hence 3 waiting lines, or there can be one service station and hence one queue. These days we mostly see one counter for airline services as this benefit the passengers and airline best. This conclusion...

Words: 2220 - Pages: 9

Free Essay

Joint Admission Control and Resource Allocation

...IEEE Globecom 2010 Workshop on Broadband Wireless Access Joint Admission Control and Resource Allocation with GoS and QoS in LTE Uplink Oscar Delgado ECE, Concordia University Montreal, Qc, H3G 1M8, Canada Email: o delgad@encs.concordia.ca Brigitte Jaumard CIISE, Concordia University Montreal, Qc, H3G 1M8, Canada Email: bjaumard@ciise.concordia.ca Abstract—In this paper, an admission control (AC) scheme is proposed for handling multiclass Grade of Service (GoS) and Quality of Service (QoS) in Uplink Long Term Evolution (LTE) systems. GoS requirement in conjunction with QoS has been seldom taken into account in previous admission control and resource allocation algorithms for LTE uplink. We propose a novel algorithm for handling the priorities while fulfilling the QoS objective of all granted requests. It corresponds to a solution that combines resource allocation and admission control properties to satisfy the GoS and QoS objectives. Call blocking probability, call outage probability, system capacity and number of effectively served requests are used as performance metrics. Numerical results show that it is possible to manage a priority scheme which satisfies the QoS constraints of all granted requests without any system capacity loss, when comparing to previous algorithms. Furthermore, the proposed AC algorithm gain, for the most sensitive traffic, can be around 20% over the reference AC algorithm. Index Terms—QoS, Priority, Admission Control, Scheduling, LTE, Uplink...

Words: 3728 - Pages: 15

Free Essay

Round Robin

...LAB 8 Task 1 Dfnalskfj;aslja;sf Asfasfasfasf Asfasf Asfasf Asfasff Gjfgjfgdhjd Dgdrd Task 2 #include<stdio.h> //#include<conio.h> main(){ int i, j, k, n, so, tq, sob, sum, swt, stat, tata, temp, count; int bt[10], bth[10], wt[10], tat[10]; float awt=0.0, atat=0.0; char new; printf("\n\n\n\n To start round robin scheduling press any key: "); k = 0; new = getchar(); system("cls"); while(k < 7){ j = 0; sob = 0; count = 0; sum = 0; swt = 0; stat = 0; tata = 0; printf("\n\n\n\t\t\t ROUND-ROBIN SCHEDULING"); printf("\n\t\t\t ======================"); printf("\n\n\n\n\n Enter number of processes: "); scanf("%d", &n); printf("\n"); for(i = 0; i < n; i++){ printf("\n Enter burst time for Process P%d: ", i+1); scanf("%d", &bt[i]); bth[i] = bt[i]; } printf("\n\n Enter time quantum: "); scanf("%d", &tq); system("cls"); printf("\n\n\n\t\t\t ROUND-ROBIN SCHEDULING"); printf("\n\t\t\t ======================"); printf("\n\n\n\n\n Time quantum: %d", tq); for(i = 0; i < n; i++){ if(bth[i] % tq == 0){ so = bth[i] / tq; } else{so = (bth[i] / tq) +1;} sob = sob + so; } int gc[sob], gcps[sob]; ...

Words: 486 - Pages: 2

Free Essay

Optimal Power Allocation and Scheduling for Two-Cell Capacity Maximization

...Optimal Power Allocation and Scheduling for Two-Cell Capacity Maximization ∗ Dept. Anders Gjendemsjø∗, David Gesbert†, Geir E. Øien∗ , and Saad G. Kiani† of Electronics and Telecom., Norwegian Univ. of Science and Technology, 7491 Trondheim, Norway, Email: {gjendems, oien}@iet.ntnu.no † Mobile Communications Department, Institute Eur´ com, e 06560 Sophia-Antipolis, France, Email: {gesbert, kiani}@eurecom.fr maximize the network capacity for the case of individual link power constraints [8] and a sum power constraint [9]. In [10] it is assumed that each base station, when it transmits, transmits with maximum power Pmax . Which base stations that should be active at each time slot is decided according to a rate maximization objective. However, no proof of optimality is given for the on/off power allocation. In [11] transmit power allocation for a downlink two-user interference channel is studied under a sum transmit power constraint and the assumption of symmetric interference. The derived power allocation depends on the level of interference; when the inference is above a certain threshold the total power is allocated to the best user. For interference less than the threshold, the available power is divided among the two users according to a water-filling principle. However, due to the sum power constraint and symmetry of interference assumption these results are not readily applicable for two-cell power allocation, where it is more reasonable to assume individual power constraints...

Words: 4991 - Pages: 20

Premium Essay

A Distributed Joint Channel-Assignment, Scheduling and Routing Algorithm for Multi-Channel Ad Hoc Wireless Networks

...A Distributed Joint Channel-Assignment, Scheduling and Routing Algorithm for Multi-Channel Ad Hoc Wireless Networks Xiaojun Lin and Shahzada Rasool School of Electrical and Computer Engineering, Purdue University West Lafayette, IN 47907, U.S.A. {linx,srasool}@ecn.purdue.edu Abstract— The capacity of ad hoc wireless networks can be substantially increased by equipping each network node with multiple radio interfaces that can operate on multiple non-overlapping channels. However, new scheduling, channelassignment, and routing algorithms are required to fully utilize the increased bandwidth in multi-channel multi-radio ad hoc networks. In this paper, we develop a fully distributed algorithm that jointly solves the channel-assignment, scheduling and routing problem. Our algorithm is an online algorithm, i.e., it does not require prior information on the offered load to the network, and can adapt automatically to the changes in the network topology and offered load. We show that our algorithm is provably efficient. That is, even compared with the optimal centralized and offline algorithm, our proposed distributed algorithm can achieve a provable fraction of the maximum system capacity. Further, the achievable fraction that we can guarantee is larger than that of some other comparable algorithms in the literature. I. I NTRODUCTION Multi-channel multi-radio ad hoc wireless networks have recently received a substantial amount of interest, especially under...

Words: 8961 - Pages: 36

Free Essay

Queuing Theory Based Approach to the Analysis of Sales Checkout at Montagu Spar Supermarket

...Great Zimbabwe University Faculty of Agriculture and Natural Sciences Department of Mathematics and Computer Science Student: Sigwadhi Teddy M149125 Research Project (HSOR 460) Proposal Presentation in partial fulfilment of BSc. 4th Year Special Honours Degree in Operations Research and Statistics Supervisor: Mr. R. Mawonike Research Topic  Queuing theory based approach to the analysis of sales checkout at Montagu Spar supermarket  Location: Avenues Area, Harare, Zimbabwe Background of the study • Zimbabwe is an important emerging country among the developing countries. • The Spar Montagu has been chosen to be the research object primarily because of its clientele which have different buying behaviors. There are a mix of customers, low to high class customers and it has been seen to provide interesting results on the busy and non busy periods. • The main purpose of this project is to study the application of queuing theory and to evaluate the parameters involved in the service unit for the sales checkout operation in Spar Montagu supermarket Background of the study continued… • Queuing theory is the theory of waiting lines and service provision • A mathematical model is to be developed to analyse the performance of the checking out service unit • Two parameters need to be determined from the data collected in the supermarket through the mathematical model to the service point. • One parameter is the customer arrival rate to the service point per hour • The other is the...

Words: 848 - Pages: 4

Free Essay

Netw320 W3 Lab

...NETW320 -- Converged Networks with Lab Lab #3 Title: IPv4 TOS and Router Queuing Objectives In this lab, you will work with an intranet for an organization that will encompass four different site locations in different cities. The subnets of these locations will be connected by a backbone IP network. The organization will be using a converged network that allows data and real-time voice traffic to traverse the same packet-switched network. The data traffic will consist of FTP (file transfer protocol) and email traffic and the voice traffic will be a VoIP (Voice over Internet Protocol) implementation. You will experiment with various router queuing policies to see how routers within a TCP/IP network can be utilized to support QoS (Quality of Service) within a converged network that is based on TCP/IP. Explanation and Background Traditional voice and data applications have been kept on separate networks. The voice traffic is confined to a circuit-switched network while data traffic is on a packet-switched network. Often, businesses keep these networks in separate rooms, or on different floors, within buildings that they own or lease (and many still do). This requires a lot of additional space and technical manpower to maintain these two distinct infrastructures. Today’s networks call for the convergence of these circuit-switching and packet-switching networks, such that voice and data traffic will traverse a common network based on packet switching. A common WAN technology...

Words: 2956 - Pages: 12

Free Essay

Netw410

...Overview In this lab, we will compare the speed and accuracy of different traffic representations: explicit traffic, background traffic, and hybrid traffic. The network used in the lab is a model of a company that provides video-on-demand services to 100 users. The company would like to introduce three classes of service for its clients: gold (ToS = 3), silver (ToS = 2) and bronze (ToS = 1). To provide differentiated treatment for the different service classes, Weighted Fair Queuing (WFQ) has been configured on the access router. In this lab, we will predict the delay for each class of service and compare the results obtained using the different traffic modeling approaches. Objectives and Methodology * Create a simple network with explicit traffic and run a simulation. * Replace explicit traffic with background traffic and rerun the simulation. * Replace background traffic with hybrid traffic and rerun the simulation. * Assess and compare the speed and accuracy of the three traffic-modeling approaches. Explanation and Background In the real world, one of the most important jobs a network manager can do is manage the traffic on the network. If the traffic doesn’t flow, the network is not exactly a credit to its operators. In addition, we now deal with many different types of traffic that are particular to certain applications and architectures, the most obvious examples being voice and video traffic. In these applications, the number one demand is for low delay...

Words: 2098 - Pages: 9

Free Essay

Bamboo House

...Queuing Theory Significance There is a very significant reason why Queuing Theory exists. Not only does it apply to a wide variety of topic, many within the business and supply chain industries, it also helps prove cause and effect. In addition to this, it provides a very logical idea of what a solution to a problem it has discovered should be. Measuring and understanding both order rate and service rate can potentially be the difference between business success and business failure. For example, if a company has too slow of a service rate, it is going to lose business because of the long wait times. On the opposite end of the spectrum, if a company focuses too much on improving its service rate instead of understanding its ratio compared to order rate, it will be misusing its very valuable resources. It is also important to have knowledge of all different types of queuing systems. Importance of Queuing Configuration As one can imagine, the importance of a queuing system configuration is very significant as well. As stated above, there are several different types of queuing systems and queuing configurations. If a business uses an improper queuing system or queuing configuration, it can suffer from one of many different negative consequences. Some examples of different types of queuing systems/queuing configurations are First Come First Served, First In First Out, Round Robin, Service in Random order, and many more. Queuing systems and configurations also vary by the number...

Words: 531 - Pages: 3

Premium Essay

Queueing Theory

...Queuing Theory Queuing Theory Waiting in lines is a social phenomenon that people face on a daily basis. Queues of people form when checking in at the airport, purchasing items at a cash register, and getting on rides in amusement parks. Waiting in lines can have both economic costs and psychological costs when customers perceive it as a negative experience. Waiting too long in line can be extremely frustrating for customers and staff. Analyzing and understanding queuing systems for service businesses involves finding and managing the best level of service that will keep customers happy and costs under control. The problem for managers in most queuing situations is the trade-off decision between adding costs of providing more rapid service against the inherent cost of waiting. To analyze a queuing system one must look at arrival characteristics including the pattern in which customers arrive at the facility, customer behaviors once they are in line, and the size of the customer population. Service characteristics, such as, the configuration of the service system and the pattern of service times must also be considered in the mathematical model (Render, Stair, Hannah, & Hail, 2015). In this paper, I will discuss the advantages and disadvantages of queuing theory for an organization in the service industry and the benefits provided by the constant service time model Queuing Systems for Service Businesses Queuing systems are put in place to serve customers in an orderly...

Words: 818 - Pages: 4

Free Essay

Project Management

...* Management optimization of bank’s queuing system * Abstract Nowadays the queue phenomenon in the bank offices is a common and troublesome issue that nearly occurs everyday in the banks of China. The rapid tempo of life makes people pay much attention to the time management, they don’t willing to spend much time on queuing and gradually lose confidence in banks. In order to improve the efficiency and the satisfy degree of customers and finally increase the profit of banks, the banks have to do something about their current queuing system. This proposal aims at analyzing the current queuing system of the China banks, finding out existing problems and carrying out some effective measures based on the previous researches, the principle of queuing and statistic method. Under the premise of less increasing the operation cost of the banks to improve the service efficiency in order to realize the win-win result of customers satisfaction and banks profit. Background Every person no matter he is young or old may have some painful experiences waiting in the banks of China. Customers still have to wait for a period of time even when they avoid the busy hours of the banks. Not alone the busy operation time or the peak period of the banks people have to wait for ages. I also had the similar experiences when I was an undergraduate student. As a student we need to pay our tuition fee through a certain bank such as Bank of China before a semester begins. However so many students...

Words: 5000 - Pages: 20

Free Essay

Queuing

...Queuing Theory Queuing Theory is generated from the service industries such as shops and retail dealers that need to pay much attention to the feeling of the customers and, at the mean time, to the cost with that the service was offered. As a retailer manager, one of the important things he or she might focus on is that the queue line length which could not be too long or too short. If the queue line is too long, the customer would be impatient and complain about the service quality the shop offers while if the shop gives too many counters to deal with the customers transaction further to reduce the length of the queue, it is definitely to increase the cost of the operation. Queue Theory is a kind of tool that could help the managers who need to analyze the queue line and estimate the cost of controlling it to understand the situation and make a decision on it. The prerequisite of Queue Theory is that the customers, services and other factors in the systems are discrete. In other words they are independent with each other since the rate of the customer coming and the rate of the service provided would not affect each other. Then these factors could meet the demand of Poisson Distribution. There are four models about Queuing Theory according to our textbook: MM1- Single-Server Queuing Model, MMS- Multiple-Server Queuing Model, MD1- Constant-Service-Time Model and Limited-Population Model. They are very useful in the different areas in the business. The first model single-server...

Words: 1374 - Pages: 6

Free Essay

Quantitative Modules:

...Quantitative Modules: I Love Coffee (ILC) illustrate a waiting-line system. The Queuing theory is a very important aspect of operations at ILC, and plays as vital tool to ILC’s operations manager. I Love Coffee drive through, implements a queuing system with its customers, having customers wait in line for the product. Characteristics of the waiting –line system at the I Love Coffee drive through. First using the arrivals or input in the systems of all the customers of (ILC) this will help build a database to better understand aspects of the ILC’s customers, such as the size or number of customers in given period of time, and certain patterns that emerge to help demonstrate a better understanding an example would be statistical distribution. With respect to the size of the population at I Love Coffee, it has an unlimited population. Patterns of customer arrivals at (ILC) show a rise in arrivals between 07:00AM – 11:00AM and 6:00PM – 12:00AM with intervals varying from 3:00 to 5:00 minutes per arrival or customer. Moreover, with regards to the waiting line characteristics ILC has a limited queuing system that can take 6 vehicles at one certain time period, more over the drive through also acts as the service facility, supplying the coffee through designated collection windows. Additionally, to have a better quantitative understanding of the arrival rate with regards to the queuing theory, a discrete Poisson distribution can be established by the following: x: 3 P(x)=...

Words: 504 - Pages: 3

Free Essay

Queiuing Theory

...Queuing Theory Most restaurants want to provide an ideal level of service wherein they could serve their customers at the least minimum time. However, as the restaurant established its name to the public, it makes a great queuing or waiting line that most of the customers do not want. Not all restaurants desire for queue since it could make confusions to them and because of their losses from the customers who go away and dissatisfied. For some time, adding chairs and tables are not enough to solve the queuing problem. In the case of Tamagoya Noodle House, they have this principle of serving the customer with their high quality ramen regardless of the number of customers. In short, they are more on the quality than the quantity; not on the profit side but rather on the quality side. But because they really want to serve more customers especially those ramen lovers who came from far places, they want to solve these queuing problems. Service time distribution Arrivals Customer 3 Customer 2 Customer 1 Service Facility Queue Fig. 1 Queuing System Configuration Assumptions of the model: Since Tamagoya Noodle House uses a Single-Channel, Single-Phase model in order to avoid confusion of customer’s order. The model we used assumes that seven conditions exist: 1. Arrivals are served on a First-in, First-out basis. Though some of customers who ordered less and or senior citizens were prioritized to be served first. 2. Every customer...

Words: 1117 - Pages: 5