QUEUING THEORY

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Kendall Notation:
- Developed to allow the key characteristics of a specific queuing model to be described in an efficient manner.
- Simple queuing models can be described by there parameters in the following general format:
1 / 2 / 3
1 / 2 / 3
Parameter 1 identifies?
Identifies the nature of the arrival process.
M = Markovian interarrival times
(Following an exponential distribution)
D = Deterministic interarrival times (Not Random)
1 / 2 / 3
Parameter 2 identifies?
Identifies the nature of the service times.
M = Markovian interarrival times
(Following an exponential distribution)
G = General service times (following a nonexponential distribution)
D = Deterministic interarrival times (Not Random)
1 / 2 / 3
Parameter 3 identifies?
Indicates the number of servers available.
M / M / 1 queue refers to a queuing model in which :
- The time between arrivals follows an exponential distribution
- The service times follow an exponential distribution
- There is one server
M / G / 3 queue refers to:
A model in which the interarrival times are assumed to be exponential, the service times follow some general distribution, & three servers are present.
Queuing models are used to:
- describe the behavior of queuing systems
- determine the level of service to provide
- evaluate alternate configurations for providing service
T or F : Random service times from an exponential distribution can assume any positive value.
TRUE
Queue :
- a waiting line
Queuing Theory :
- body of knowledge dealing with waiting lines
M / M / s model is appropriate for analyzing queuing problems when these specific assumptions are met:
Assumptions:
- There are s servers, where s is a positive integer
- Arrivals follow a Poisson distribution and occur at an average rate of l per time period.
- Each server provides service at an average rate of u per time period, and actual service times follow an exponential distribution.
- Arrivals wait in a single FIFO queue and are serviced by the first available server.
- l < s u
Results for the M / M /s models assume that:
The size or capacity of the waiting area is infinite, so that all arrivals to the system join the queue and wait for service.
What does the final assumption indicate?
l (symbol for arrival rate) < s u
That the total service capacity of the system, s u , must be strictly greater than the rate at which arrivals occur l (symbol for arrival rate).
If the arrival rate exceeds the system's total service capacity, then :
The system would fill up over time, and the queue would become infinitely long.
** queue will also become infinitely long even if the average arrival rate is equal to the average service rate s u
Will there be times when the servers are idle ?!
- Yes, and this idle time is lost forever.
- The servers will not be able to make up for this at other times when the demand for service is heavy.
Infinite Queue :
Queue which continues to expand
(calling units are coming faster than server can handle them).
Finite Queue Length :
The size or capacity of the waiting area has a restriction
In some problems, the amount of waiting area is limited.
This means that rather than wait for service, units will balk.
Balk :
Refers to an arrival that does not join the queue because the queue is full or too long.
M / M / s model with finite population :
* these queuing models have a finite arrival (or calling) population
* the average arrival rate for the system changes depending on the number of customers in the queue
M / M / s model with finite population is appropriate for analyzing queuing problems where the following assumptions are met:
- there are s servers, where s is a positive integer
- there are N potential customers in the arrival population
- the arrival pattern of each customer follows a Poisson distribution with a mean arrival rate of l per time period
- each server provides service at an average rate of u per time period, and actual service times follow an exponential distribution
- arrivals wait in a single FIFO queue and are serviced by the first available server

** note the avg arrival rate for this model is defined in terms of the rate at which each customer arrives
M / G / 1 Queuing Model :
- Enables us to analyze queuing problems in which service times cannot be modeled accurately using an exponential distribution
- this queuing model is remarkable because it can be used to compute the operating characteristics for any one-server queuing system where arrivals follow a Poisson distribution and the mean u and standard deviation o of the service time are known.
^^ essentially can be used when service times are random with known mean and standard deviation
Renege :
- leave a queue before being served
Jockey :
- switch from one queue to another
Typical operating characteristics of interest include:
U
Utilization factor, % of time that all servers are busy
P(0)
- Probability that there are no zero units in the system
Lq
- Average number of units in line waiting for service
L
- Average number of units in the system (in line & being served)
Wq
- Average time a unit spends in line waiting for service
W
- Average time a unit spends in the system (in line & being served)
Pw
- Probability that an arriving unit has to wait for service
Pn
- Probability of n units in the system
Arrival Rate :
- the manner in which customers arrive at the system for service
- model the arrival process in a queuing system as a Poisson random variable
Interarrival Time:
- random amount of time that is likely to transpire between arriving calls
- follows an exponential probability distribution with mean 1 / l
^^ if the # of arrivals in a given period follows a Poisson distribution with mean l
Exponential Distribution :
- plays a key role in queuing models
- one of the few probability distributions that exhibits the memory-less property
Markov:
First to recognize the memory less property of certain random variables; thus memoryless property is referred to as the Markov or Markovian property
Queue Time:
Amount of time a customer spends waiting in line for service to being
Service Time:
Amount of time a customer spends at a service facility after the actual performance of service begins
Queue time VS Service time
SERVICE TIME DOES NOT INCLUDE QUEUE TIME!
Service Rate :
- denoted by u
- representing the average number of customers (or jobs) that can be served per time period
- average service time per customer is 1 / u time periods
- variance of the service time per customer is (1/u)^2 time periods
What Operating Characteristic will give you the "Probability that there are no trucks in the system" ?
P(0)
What Operating Characteristic will give you the "average number of trucks waiting for service"?
Lq
What Operating Characteristic will give you the "average time a truck waits for the loading/unloading service to begin"?
Wq
What Operating Characteristic will give you the "probability a new arrival will have to wait"?
- Pw
- Probability that a customer waits.
L - Lq =
- Average # of number of units actually being serviced
W - Wq =
- Average time it takes a unit to be serviced
With respect to farther improving profit, would you make any other recommendations to The Three Little Pigs regarding the operation of their checkout counters?
I would recommend that Hansel and Gretel find something for their employees to do when they are not busy operating their checkout stand.
If all the operating characteristics of a M / M / S = #DIV/0 then what does this mean for the queue?
This means the queue of customer/trucks/units of whatever would increase endlessly.
If the Lq, L, W, & Wq of a M/M/S = a negative number then what does this mean for the queue?
It means we have an infinite queue.
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