# Birth Death Model: M/M/C Queueing System

• MCooltA
In summary, the conversation discusses two types of problems related to M/M/C queueing systems. The first problem involves a constant arrival rate of λ=10, service rate of μ=5.122, and 2 servers (C=2). The steady state probabilities Pn are determined using the formula p_{n} = (\rho^{n} / n!)(1/S) for n ≤ c and p_{n} = (\rho^{c} / c!)(\rho/c)^{n-c} (1/S) for n ≥ c+1, with S = \sum_{n=0}^{c} \rho^{n}/n! + (\rho^{c}/c!)((\rho/c)/(
MCooltA

## Homework Statement

So for M/M/C queueing systems, I have two types of problems that I have been asked to deal with.

For the first question I am given in my notes, is where the arrival rate is constant λ=10 and service rate is μ=5.122 with 2 servers, i.e. C=2. For this question, I have used the formula to determine the steady state probabilities Pn, which denotes the probability of having n customers in the system.

$$p_{n} = (\rho^{n} / n!)(1/S), n ≤ c$$
$$p_{n} = (\rho^{c} / c!)(\rho/c)^{n-c} (1/S), n ≥ c+1$$

where $$S = \sum_{n=0}^{c} \rho^{n}/n! + (\rho^{c}/c!)((\rho/c)/(1-(\rho/c))$$

On another question, I am given the arrival rate λ=4/(n+1) where n is the number of customers in the system, and μ(the service rate) =2, C(servers) =3 with the system being full once there are 4 customers in the system.

To calculate the steady state probabilities for this question, we have used the formula;

$$S=1 + λ_{0}/μ_{1} + λ_{1}/μ_{2} + ... = 1/p_{0}$$

$$p_{n} = (λ_{0}λ_{1} + λ_{2}... λ_{n-1})/(μ_{1}μ_{2}μ_{3}...μ_{n})$$

I am having difficulty understanding which formula I should use, depending on the problem. Is the first formula only required when I have an M/M/C queue and the arrival rate is constant.

MCooltA said:

## Homework Statement

So for M/M/C queueing systems, I have two types of problems that I have been asked to deal with.

For the first question I am given in my notes, is where the arrival rate is constant λ=10 and service rate is μ=5.122 with 2 servers, i.e. C=2. For this question, I have used the formula to determine the steady state probabilities Pn, which denotes the probability of having n customers in the system.

$$p_{n} = (\rho^{n} / n!)(1/S), n ≤ c$$
$$p_{n} = (\rho^{c} / c!)(\rho/c)^{n-c} (1/S), n ≥ c+1$$

where $$S = \sum_{n=0}^{c} \rho^{n}/n! + (\rho^{c}/c!)((\rho/c)/(1-(\rho/c))$$

On another question, I am given the arrival rate λ=4/(n+1) where n is the number of customers in the system, and μ(the service rate) =2, C(servers) =3 with the system being full once there are 4 customers in the system.

To calculate the steady state probabilities for this question, we have used the formula;

$$S=1 + λ_{0}/μ_{1} + λ_{1}/μ_{2} + ... = 1/p_{0}$$

$$p_{n} = (λ_{0}λ_{1} + λ_{2}... λ_{n-1})/(μ_{1}μ_{2}μ_{3}...μ_{n})$$

I am having difficulty understanding which formula I should use, depending on the problem. Is the first formula only required when I have an M/M/C queue and the arrival rate is constant.

Right. So that means you cannot use the first formula in the second problem, because: (i) the arrival rate is not constant; and (ii) the queue capacity is finite. (Note: the first formula does not apply either if the queue capacity is finite. There are fancier formulas you can use instead.)

I cannot make any sense of your pn formula in the second case; it looks wrong to me.

It is a mistake to use canned formulas for problems like the second one (unless it, too, happened to be done in your notes). Instead, start from first principles. (1) You have a birth-death process. (2) Draw the transition diagram. (3) Write down in detail, one-by-one, the steady-state equations. (4) Solve the equations [NOT difficul!].

## 1. What is a Birth Death Model?

A Birth Death Model is a mathematical model used to study systems that involve the arrival and departure of entities over time. It is commonly used in queueing systems to understand the behavior of waiting lines and service processes.

## 2. What is an M/M/C Queueing System?

An M/M/C Queueing System is a type of queueing system that follows the Markovian Arrival Process (M), Markovian Service Process (M), and has a C number of servers. This means that the arrival of entities is random, the service times are exponentially distributed, and there are a fixed number of servers serving the queue.

## 3. How is the Birth Death Model used in M/M/C Queueing Systems?

The Birth Death Model is used in M/M/C Queueing Systems to analyze the expected waiting time, queue length, and utilization of the servers. It helps in predicting the performance of the system and identifying areas for improvement.

## 4. What are the assumptions made in the Birth Death Model for M/M/C Queueing Systems?

The assumptions made in the Birth Death Model for M/M/C Queueing Systems include the system being in a steady state, arrival and service processes being independent and exponentially distributed, and there being no balking or reneging in the queue.

## 5. What are the practical applications of the Birth Death Model for M/M/C Queueing Systems?

The Birth Death Model for M/M/C Queueing Systems has practical applications in various fields, such as transportation, telecommunications, healthcare, and manufacturing. It can be used to optimize resource allocation, improve customer satisfaction, and reduce waiting times in queues.

• Calculus and Beyond Homework Help
Replies
3
Views
717
• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
18
Views
2K
• Calculus and Beyond Homework Help
Replies
3
Views
730
• Calculus and Beyond Homework Help
Replies
3
Views
736
• Calculus and Beyond Homework Help
Replies
2
Views
838
• Calculus and Beyond Homework Help
Replies
1
Views
1K