“Queuing Theory”

Posted by husnulkhuluq | Mathematics | Thursday 16 December 2010 07:36

Queuing Theory is first declared by Agner Kraup Erlang in 1909 with the paper published entitle “Queuing Theory”. The idea comes from his observation toward the limited capacity problem of service telephone to service consumer demand in a certain time. There are some examples with some unique pattern problems like traffic, bank service on the month or every week. The unique pattern we mean here is the lately service which is caused by the service demand rate which is more than the capability of facility to serve. However, the unique of the pattern shows that there are many processes to recognize together with their assumptions.

A. Basic Concept of Model
Basic Purpose
Queuing models aims to minimize a direct cost to supply the service and individual cost for they are waiting for the service. The different between the number of demands and the capability of service facility causes two consequences, are queuing and empty capacity. A long queue implies waited-line that causes an opportunity cost to the costumer. Another issue, too many service facilities affect meaningless facilities. That’s why they both should be minimized.

System and Parameters
There are four dominant factors toward the system approach such as, System Boundaries, Input, Process, and Output. System boundary gives us a limitation of the area observing in queue. Input implies the people need to get a service from a supplier facility. The process means the service activity itself while output states the customer who is completely served.

There are two variables that influence waiting line shape that is arrival rate (λ) and service rate (μ). If arrival rate is greater than the service rate then there is waiting line. Therefore, we should assume λ > μ to guarantee that process do not stop because of a very high demand.
Next, we will show the simulation of queuing basic system.

Arrival Rate (λ) and Poisson Process
Let’s think about the observation of A. K. Erlang in Copenhagen Telephone, the pattern of costumers demand on continuous of time can be divided into several fixed intervals. In this case, the demand of the costumer is distributed randomly in each fixed intervals. It is known as Poisson Process.

From the example, there are 10 costumers came in 06.00 – 10.00. But, it is different from what happened in the other intervals. This is an example of phenomena observed by A.K. Erlang and followed by Poisson process which is often occur in any queuing cases. In this case, the assumption as follows:
1. Arrival costumer is randomly
2. Arrival costumer in each time interval didn’t influence each other.

From the example, the time interval is divided into four fixed intervals. If I is the amount of time interval, then

Where, Ii is the i-th interval

This case shows I1 = 1 interval with 6 arrivals; I2 = 1 interval with 1 arrival; I3 = 1 interval with 0 arrival; and I4 = 1 interval with 3 arrivals. Hence I = 4. If N symbolizes the number of costumers coming during the intervals, and there is Ki customers in interval Ii, hence the number of costumer during I is:

In this case, N = 6+1+0+3 = 10.
So the costumers come randomly to every similar interval. If every interval divided into n subintervals then with the same process and assumption, the time interval can be stated as Poisson distribution. Therefore, the arrival rate of the costumers in each fixed interval can be estimated as

Service Rate (μ)
Service rate is the average time needed to serve a customer. If the capacity of service facility can serve 4 costumers per hours, then the service rate is µ = 4/hours, and the service time (time needed to serve each costumer) is 1/µ = 15 minutes. Therefore, we know that the service time satisfies negative exponential distributions while the service rate satisfies Poisson distribution.

The Length of Waited-Line
In fact, the length of waited line bounded by the space capacity to wait before getting. But, finite waited line capacity need special development model. That’s why in common model discussion; it’s assumed infinite.

Queuing Rule
Waited line or queuing is formed when the service of facility is busy servicing the costumer, so the other costumers have to wait. If the service is finished, then the first customer in queuing line is will be first served. The rule is clear, first come first served.

Equilibrium Concept in System
Queuing model use equilibrium concept or balancing the number of costumers in system as a ground to develop the model. If there is N costumers in a system then a costumer is getting out of system after being served hence the number of costumer in the system become N-1, then a costumer is coming into the system hence the number of costumer become N.
If, λ = arrival rate
µ = service rate
Pn = probability of n costumer in system
Pn-1 = probability of n-1 costumer in system

THEN the concept of equilibrium can be written as OR

If λ as arrival rate and µ as service rate with λ > µ as assumption, then busy rate of the system denoted by ρ is

“Queuing Theory – Model Configuration”

Posted by husnulkhuluq | Mathematics | Wednesday 8 December 2010 16:23

As we can see in reality, queuing happened in some public services has various forms. It might be caused by the number of services they are going to have or the number of people is in the queuing. For this case, we can analyze four items, are:
 Length of System (Ls)
 Time Spent in System (Ts)
 Length of Queue (Lq)
 Time Waiting in the Queue (Tq)

1. Length of System (Ls)
The length of system or Ls is the number of customers is in system, including they are still in queuing or in serving. Ls is influenced by two parameters are arrival rate (λ) and service rate (μ). Another thing to concern, the busier the system is, the lower the probability of empty system, and vice versa. That’s why; it is logically to define Ls as the comparison of the business rate (λ/μ) with the probability of empty service.

2. Time Spent in System (Ts)
As the definition of Ls, we can guess to define the time of system or Ts as the total time during the customer entering the queue until totally serviced. Ts will of course is be influenced by Ls and arrival rate or λ.
Therefore, Ls = λ . Ts
Let’s compare it with the previous formula we got,

3. Time Waiting in the Queue (Tq)
Time waiting in the queue or Tq is defined as the total time during the costumer is in queue (entering system until having service). So, the difference from Ts itself is only the service time (1/μ).
Therefore,

Let’s substitute the formula of Ts into that equation,

4. Length of Queue (Lq)
The length of queue or Lq is influenced by arrival rate (λ) and the time waiting in the queue (Tq). Thus,

Here, we have also four configuration model of queuing that we always meet in our daily life, such as:
 Single Channel Single Phase
 Multi Channel Single Phase
 Single Channel Multi Phase
 Multi Channel Multi Phase

In this case, we define channel as the number of servicer in every activity to serve such as the number of cashier available in the supermarket, the number of ATM machine in a place, and many others. Phase itself is defined as the number of services that the customers should have to complete servicing.
1. Single Channel Single Phase
This type is the simplest model configuration of queuing. It only consists of one row queuing and a kind of service to have. In daily life, we can see it in a single ATM machine or in one-locket ticketing service. We can also draw it as the following:

2. Multi Channel Single Phase
In daily life, this model can be seen in some cashiers in supermarket. Where there are many cashiers in that place but we should only queue to one cashier to finish paying.
The main concept of this configuration is;
No queuing if n≤k or the number of costumer is at most equals to the number of facilities serving.
There is a queue if n>k or the number of costumer to serve is greater than the number of facilities serving.

3. Single Channel Multi Phase
Single channel multi phase describes another queue model configuration which involves more than one kind processes and only one server in each serving process. We can see this queuing model in police office when we want to make a driving license (SIM). Where there are some services to have such as healthy testing service, photo service, etc.

4. Multi Channel Multi Phase
Multi channel multi phase is the most complicated queuing model. It involves many activities and many services too in each activity we have. This model can be seen in SNM-PTN registration where there are many activities we face such as taking form, paying service, giving back the form, etc and there are also many servers in each activity we should complete.

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