Question

Discuss the following queueing situations in terms of the characteristics (1. arrival pattern of customers, 2. service pattern of servers, 3. queue discipline, 4.system capacity, 5. number of service channels, and 6. number of service stages.

(a) Tourists wishing a guided tour of the White House.
(b) Processing of programs coming from a number of independent sources on a local area network into a central computer.

Answer 1

1. Arrival pattern of customers: In queuing the arrival process is usually stochastic. As a result itis necessary to determine the probability distribution of the interarrival times (times betweensuccessive customer arrivals) as well. Also customers can arrive in individually orsimultaneously (batch or bulk arrivals).

2. Service pattern of customers: As in arrivals, a probability distribution is needed for describingthe sequence of customer service time. Service may also be single or batch. The serviceprocess may depend on the number of customers waiting in queue for service. In this it iscalled state dependent service.

3. Queue discipline: Queue discipline refers to the manner in which customers are selected forservice when a queue has formed. The default is FCFS i.e. is first come first served. Someothers are LCFS (last come first served), RSS (random service selection) i.e. selection forservice in random order independent of the time of arrival and there are other priority systemswhere customers are given priorities upon entering the system, ones with higher priority areselected first.

4. System capacity: A queuing system can be finite or infinite. In certain queuing process thereis a limitation on the length of the queue i.e. customers are not allowed to enter if the queuehas reached a certain length. These are called finite queuing systems. If there is no restrictionon the length of the queue then it is called an infinite queuing system.

5. Number of service channels: A queuing system can be single or a multiserver system. In amultiserver queuing system there are several parallel servers running to serve a single line.

6.Number of service stages: A queuing system may have only a single stage of service. But as an example of a multistage queuing system consider the physical examination procedure, where each patient proceeds through various stages of medical examination, like throat check up, eye test, blood test etc.

Answer 2

1. Arrival pattern of customers: In queuing the arrival process is usually stochastic. As a result itis necessary to determine the probability distribution of the interarrival times (times betweensuccessive customer arrivals) as well. Also customers can arrive in individually orsimultaneously (batch or bulk arrivals).
2. Service pattern of customers: As in arrivals, a probability distribution is needed for describingthe sequence of customer service time. Service may also be single or batch. The serviceprocess may depend on the number of customers waiting in queue for service. In this it iscalled state dependent service.
3. Queue discipline: Queue discipline refers to the manner in which customers are selected forservice when a queue has formed. The default is FCFS i.e. is first come first served. Someothers are LCFS (last come first served), RSS (random service selection) i.e. selection forservice in random order independent of the time of arrival and there are other priority systemswhere customers are given priorities upon entering the system, ones with higher priority areselected first.
4. System capacity: A queuing system can be finite or infinite. In certain queuing process thereis a limitation on the length of the queue i.e. customers are not allowed to enter if the queuehas reached a certain length. These are called finite queuing systems. If there is no restrictionon the length of the queue then it is called an infinite queuing system.
5. Number of service channels: A queuing system can be single or a multiserver system. In amultiserver queuing system there are several parallel servers running to serve a single line.
6.Number of service stages: A queuing system may have only a single stage of service. But as an example of a multistage queuing system consider the physical examination procedure, where each patient proceeds through various stages of medical examination, like throat check up, eye test, blood test etc.

Answer 3

Construction and analysis
Queueing models are generally constructed to represent the steady state of a queueing system, that is, the typical, long run or average state of the system. As a consequence, these are stochastic models that represent the probability that a queueing system will be found in a particular configuration or state.
A general procedure for constructing and analysing such queueing models is:
Identify the parameters of the system, such as the arrival rate, service time, queue capacity, and perhaps draw a diagram of the system.
Identify the system states. (A state will generally represent the integer number of customers, people, jobs, calls, messages, etc. in the system and may or may not be limited.)
Draw a state transition diagram that represents the possible system states and identify the rates to enter and leave each state. This diagram is a representation of a Markov chain.
Because the state transition diagram represents the steady state situation between state there is a balanced flow between states so the probabilities of being in adjacent states can be related mathematically in terms of the arrival and service rates and state probabilities.
Express all the state probabilities in terms of the empty state probability, using the inter-state transition relationships.
Determine the empty state probability by using the fact that all state probabilities always sum to 1.
Whereas specific problems that have small finite state models can often be analysed numerically, analysis of more general models, using calculus, yields useful formulae that can be applied to whole classes of problems.
[edit]Single-server queue
Single-server queues are, perhaps, the most commonly encountered queueing situation in real life. One encounters a queue with a single server in many situations, including business (e.g. sales clerk), industry (e.g. a production line), transport (e.g. a queues that the customer can select from.) Consequently, being able to model and analyse a single server queue's behaviour is a particularly useful thing to do.
[edit]Poisson arrivals and service
M/M/1// represents a single server that has unlimited queue capacity and infinite calling population, both arrivals and service are Poisson (or random) processes, meaning the statistical distribution of both the inter-arrival times and the service times follow the exponential distribution. Because of the mathematical nature of the exponential distribution, a number of quite simple relationships are able to be derived for several performance measures based on knowing the arrival rate and service rate.
This is fortunate because an M/M/1 queuing model can be used to approximate many queuing situations.
[edit]Poisson arrivals and general service
M/G/1// represents a single server that has unlimited queue capacity and infinite calling population, while the arrival is still Poisson process, meaning the statistical distribution of the inter-arrival times still follow the exponential distribution, the distribution of the service time does not. The distribution of the service time may follow any general statistical distribution, not just exponential. Relationships are still able to be derived for a (limited) number of performance measures if one knows the arrival rate and the mean and variance of the service rate. However the derivations are generally more complex and difficult.
A number of special cases of M/G/1 provide specific solutions that give broad insights into the best model to choose for specific queueing situations because they permit the comparison of those solutions to the performance of an M/M/1 model.
[edit]Multiple-servers queue
Multiple (identical)-servers queue situations are frequently encountered in telecommunications or a customer service environment. When modelling these situations care is needed to ensure that it is a multiple servers queue, not a network of single server queues, because results may differ depending on how the queuing model behaves.
One observational insight provided by comparing queuing models is that a single queue with multiple servers performs better than each server having their own queue and that a single large pool of servers performs better than two or more smaller pools, even though there are the same total number of servers in the system.
One simple example to prove the above fact is as follows: Consider a system having 8 input lines, single queue and 8 servers.The output line has a capacity of 64 kbit/s. Considering the arrival rate at each input as 2 packets/s. So, the total arrival rate is 16 packets/s. With an average of 2000 bits per packet, the service rate is 64 kbit/s/2000b = 32 packets/s. Hence, the average response time of the system is 1/(µ - ?) = 1/(32 - 16) = 0.0625 sec. Now, consider a second system with 8 queues, one for each server. Each of the 8 output lines has a capacity of 8 kbit/s. The calculation yields the response time as 1/(µ - ?) = 1/(4 - 2) = 0.5 sec. And the average waiting time in the queue in the first case is ?/(1 - ?)µ = 0.03125, while in the second case is 0.25.
[edit]Infinitely many servers
While never exactly encountered in reality, an infinite-servers (e.g. M/M/) model is a convenient theoretical model for situations that involve storage or delay, such as parking lots, warehouses and even atomic transitions. In these models there is no queue, as such, instead each arriving customer receives service. When viewed from the outside, the model appears to delay or store each customer for some time.

Answer 4

(a)
Tourists wishing a guided tour of the White House.
• Arrival pattern of customers:
The arrival pattern of customers in the case of tourists wishing a guided tour is dynamic arrival process. Number of tourists arriving will vary. So depending on the arrival rate, the appointments are scheduled and may be changed.
• Service pattern of servers:
1. Configuration of the service system: Single server single queue model is to be configured. Tourist waits till the guide is available to provide service.
2. Speed of service: The service rate and service time are to be estimated. The number of tourists that can be serviced in a particular time and the time required to service each customer are specified.
• Queue discipline: Priority service queue discipline may be suited as the priorities of the tourists vary.
• System capacity: Based on the flow of tourists visiting and the capability of servicing, it must be decided that how many tourists are to be serviced per day or in a time period.
• Number of channels: This is a single channel system as a guided tour is a single service system.
• Number of service stages: One service stage may service the customers. In one stage the task can be completed.


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