Question

Are these birth and death models? And if so, what is λn and what is μn?

1. M/M/1 queue with balking: an M/M/1 system except for a customer that finds n

   others in the system upon its arrival will only join the system with probability αn

2. M/M/1/K queuing system: This is a M/M/1 queuing system with finite capacity K,

   where the system can only have K individuals at maximum (arrivals when the system

   is full are turned away)

3. M/M/s queuing system: M/M/1 queuing system, except there are now s servers

   instead of just 1

Answer 1

When arrival and departure rate depends upon the number of customers in the system and queueing system use to be exponential, it is known as birth and death queueing models. Here, μn denotes the departure rate and λn denotes arrival rate when there is a number(n) customers present in the system. If there are n customers in the system, the time until next arrival is exponential with rate n and independent of the next departure time. Yes, these are birth and death models.

M/M/1 queue with balking: M/M/1 queue is the simplest of the queueing model. Customers are supposed to join n others already present in the system will only join the system with αn probability. Probability (1-αn) balks at joining the system in the queueing system. This system will be birth and death model with :

λn = λαn ,where n≥ 0

μn = μ, where n≥ 1

M/M/1/K queueing system: This model is mainly used for the research purpose.  The capacity of the system to be considered is known and it is assumed to be k. If there is, the finite waiting room of size K, it is known as the M/M/1/K queue. We consider that the server is busy if there is at least one customer in the process. The M/M/1/K queueing system can be modeled as death and birth queueing system  with below-assumed parameters:

λn = λ, if 0 ≤ n<k

λn = 0 , if n ≥ k

  μn = μ, where n≥ 1

M/M/s queueing system: Customers use to arrive according to a Poisson process with arrival rate λ. The service time of customers is exponentially distributed with parameter µ. There are s servers, which is serving customers at the order of arrival. this may be model after the well-known queue model, where s is the number of systems, here I take the value from I to s or any value that may suit the analysis.

Stability condition:

λ < s · µ

ρ = λ /s · µ < 1.

The process {X(t), t ≥ 0}, the number of customers in the system at time t, and this system is written as birth and death form.