Queuing Theory represents the body of knowledge dealing with waiting lines Most queuing problems focus on determining the level of service that a company should provide Queuing Theory Queuing Systems Configurations ID: 132544
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Slide1
Queuing Theory
Queuing Theory represents the body of knowledge dealing with waiting lines.Most queuing problems focus on determining the level of service that a company should provide.Slide2
Queuing Theory
Queuing Systems ConfigurationsSlide3
Queuing Theory
Generation of CustomersInfinite vs. Finite calling populationHomogeneity of the calling populationIndividual vs. Batch arrivals
Deterministic vs. Stochastic arrivalsQueuing of CustomersSingle vs. Multiple servers
Finite vs. Infinite queues
Characteristics of a Queuing ProcessSlide4
Queuing Theory
FIFO vs. LIFO disciplinesPriority rulesServicing the CustomersDeterministic vs. Stochastic service time
Individual vs. Batch Processing
Characteristics of a Queuing ProcessSlide5
Queuing Theory
Generation of Customers Poisson probability distribution
‘x’ represents the number of arrivals in a specific time period.
‘’ represents the ‘arrival rate’, that is, the average number of arrivals per time period.
Characteristics of a Queuing ProcessSlide6
Queuing Theory
The time between arrivals is known as the interarrival time. If the number of arrivals in a given period follows a Poisson distribution, with mean
, the interarrival times follow an
exponential probability distribution with mean 1/The exponential distribution exhibit the
memoryless
property. An arrival process is memoryless if the time until the next arrival occurs does not depend on how much time has elapsed since the last arrival.
Arrival RateSlide7
Queuing Theory
Arrival RateSlide8
Queuing Theory
Queue time is the amount of time a customer spends waiting in line for service to begin.
Service time is the amount of time a customer spends at a service facility once the actual performance of service begins.
Service time is often model as an exponential random variable
Service RateSlide9
Queuing Theory
The service rate, denoted by
, represents the average number of customers that can be served per time period. The average service time per customer is 1/
time periods.
Service RateSlide10
Queuing Theory
1/2/3The first characteristic identifies the nature of the arrival process using the following standard abbreviations:
M = Markovian interarrival times (following an exponential distribution)
D = Deterministic interarrival times (not random)
Kendall NotationSlide11
Queuing Theory
The second characteristic identifies the nature of the service times using the following standard abbreviations:M = Markovian service times
G = General service times (following a non-exponential distribution)D = Deterministic service times (not random)
The third characteristic indicates the number of servers available.
Kendall NotationSlide12
Queuing Theory
U - Utilization factor, or the percentage of time that all servers are busy.
P0 - Probability that there are no units in the system.
Lq - Average number of units in line waiting for service
L
- Average number of units in the system (in line and being served)
W
q
- Average time a unit spends in line waiting for service
T
- Actual time a unit spends in the queue
W
- Average time a unit spends in the system (in line and being served)
P
w
- Probability that an arriving unit has to wait for service
P
n
- Probability of n units in the system
Operating CharacteristicsSlide13
Queuing Theory
There are s servers in the system, where s is a positive integerArrivals follow a Poisson distribution and occur at an average rate of
per time period
Each server provides service at an average rate of per time period, and actual service times follow an exponential distributionArrivals wait in a single FIFO queue and are serviced by the first available server
< s
The M/M/s ModelSlide14
Queuing Theory
Formulas describing the M/M/s ModelSlide15
Queuing Theory
Formulas describing the M/M/s ModelSlide16
Queuing Theory
Q.xlsSlide17
Queuing Theory
Case Problem (A) p. 140Slide18
Queuing Theory
Case Problem (cont.)Slide19
Queuing Theory
Case Problem (cont.)Slide20
Queuing Theory
Case Problem (cont.)Slide21
Queuing Theory
Case Problem (cont.)Slide22
Queuing Theory
Finite Queue Model
Case Problem (cont.)Slide23
Queuing Theory
Finite Queue Model
Case Problem (cont.)Slide24
Queuing Theory
Case Problem (cont.)Slide25
Queuing Theory
Case Problem (cont.)