# Birth and death process and Little's law

• I
• LogarithmLuke
In summary, the conversation discusses a model with two possible states (susceptible and infected) and assumptions of independence and exponential distributions for time until next infection and duration of infections. The birth and death rates for this birth and death process are determined to be 𝜆 and 𝜇, respectively, and the total population size is 5.26 million individuals. Little's Law is then used to calculate the average treatment time for individuals with complications from a cold, with a given probability of 1% and a hospital capacity of 2000 individuals. The long term average arrival rate is also mentioned as a way to calculate this treatment time.
LogarithmLuke
Assume that an individual only has two possible states: susceptible (S) and infected (I). Further, assume that the individuals in the population are independent, and that for each susceptible individual the time until the next infection follows an exponential distribution with expected value 1/λ = 100 days and that the durations of the infections follow independent exponential distributions with expected values 1/μ = 7 days.

a) Let Y (t) denote the number of infected individuals in the population at time t measured in days. Specify the birth and death rates of this birth and death process.

The total population contains 5.26 million individuals.

b) Assume the stochastic process {Y (t) : t ≥ 0} has reached its stationary distribution. For each infection, there is a probability of 1% that the infection will result in serious compli- cations that requires hospitalization. On average, the hospitals only have capacity to handle 2000 individuals with complications from a cold. Use Little’s law to calculate the average treatment time required to not exceed the capacity.So for a) the birth rates should decrease with an increase in the amount of infected individuals, however the overall population size remains the same in this model, so I am thinking the birth rate is just 𝜆 and the death rate simply 𝜇? Or should the death rate be 𝜇𝑖 and birth rate 𝜆/𝑖 where 𝑖 is the number of infected individuals?

For b) I am not sure what I need to calculate. From what I see we can just plug straight into Little's law 𝐿=𝜆𝑊 and put 𝐿=2000
and use the 𝜆 given to solve for 𝑊, however the answer obtained that way makes no sense.

LogarithmLuke said:
Assume that an individual only has two possible states: susceptible (S) and infected (I). Further, assume that the individuals in the population are independent, and that for each susceptible individual the time until the next infection follows an exponential distribution with expected value 1/λ = 100 days and that the durations of the infections follow independent exponential distributions with expected values 1/μ = 7 days.

a) Let Y (t) denote the number of infected individuals in the population at time t measured in days.
...

So for a) the birth rates should decrease with an increase in the amount of infected individuals, however the overall population size remains the same in this model, so I am thinking the birth rate is just 𝜆 and the death rate simply 𝜇? Or should the death rate be 𝜇𝑖 and birth rate 𝜆/𝑖 where 𝑖 is the number of infected individuals?

For b) I am not sure what I need to calculate. From what I see we can just plug straight into Little's law 𝐿=𝜆𝑊 and put 𝐿=2000
and use the 𝜆 given to solve for 𝑊, however the answer obtained that way makes no sense.

I'm not following your line of thinking here . I don't follow how you got ii in the denominator with ##\lambda##. If it isn't guessing -- how do you justify these things?

The right approach is to start with a very simple case and figure out what happens when you convolve two independent exponentials...
- - - -
I do see ##n=5.26## million and

##\lambda \cdot (n−i)## as the birth/infection rate and
##\mu \cdot i## as the death/ get healthy rate

How can these things be justified? Also have you drawn a picture of the underlying jump chain?

As for (b) what is your understanding of Little's Law?

Last edited:
The reason i proposed $$\frac{\lambda}{i}$$is because the birth rate becomes lower as more people get infected. But with that reasoning $$\lambda * (n-i)$$for the birth rate, and $$\mu i$$ for the death rate as you propose also makes sense. Yes I have sketched the jump chain, the states can either go from i to i +1 or i-1 (birth and death process).

My understanding of Little's law is that the long term average number L of customers in a stationary system is equal to the long term average arrival rate times the average serving time W.

So I am guessing I have to find the the long term average arrival rate? I was given the hint that I could use the long-run mean fraction of time per year that a single individual has a cold (which I calculated to be 7/107) instead of calculating the stationary distribution for Y(t).

LogarithmLuke said:
The reason i proposed $$\frac{\lambda}{i}$$is because the birth rate becomes lower as more people get infected. But with that reasoning $$\lambda * (n-i)$$for the birth rate, and $$\mu i$$ for the death rate as you propose also makes sense. Yes I have sketched the jump chain, the states can either go from i to i +1 or i-1 (birth and death process).
I may be overly attuned to linearity and non-linearities but here's what doesn't sit well with me.

You are right that the birth rate is bigger when small number of infected people and smaller when large number of infected.

But how is it calculated? If we ignore the getting healthy rate, which by memorylessness, we can (for now),

then the time until next infection at state i is given by the first arrival of ##n-i## poisson processes. That is there are ##(n-i)## people who aren't sick, but 'the clock is ticking' as to time until when they get sick... If you know how to combine (superpose) and split Poisson processes you should recognize that this merged process has parameter ##(n-i)\lambda## -- i.e. the rate parameter scales linearly with the number of processes. I'm not sure what text you are using, but the book by Blitzstein and Hwang does a good job treating this in chapter 13. It is freely available here:

https://projects.iq.harvard.edu/stat110/home

Little's Law is also in chapter 13...

(note: this seems to indicate that your problems are undergrad level homework, and hence may be better located in the homework forums)

LogarithmLuke said:
My understanding of Little's law is that the long term average number L of customers in a stationary system is equal to the long term average arrival rate times the average serving time W.

So I am guessing I have to find the the long term average arrival rate? I was given the hint that I could use the long-run mean fraction of time per year that a single individual has a cold (which I calculated to be 7/107) instead of calculating the stationary distribution for Y(t).
give it a shot and show me your work, and what you end up with here. I think if you explore it a bit you'll end up with a correct answer

So if the birth rate is $$\lambda(n-i)$$ and we know that approx 0.065 of the population are infected. So our arrival rate becomes $$\lambda(5.26*10^6 - 0.065*5.26*10^6)*0.01$$ since only 1% of arrivals require hospitalization. This gives $$W = \approx 4 days$$ which seems like a reasonable answer?

LogarithmLuke said:
So if the birth rate is $$\lambda(n-i)$$ and we know that approx 0.065 of the population are infected
I'm not sure where this bold section is coming from -- they weren't stated in the original problem and you haven't shown any work.

LogarithmLuke said:
b) Assume the stochastic process {Y (t) : t ≥ 0} has reached its stationary distribution. For each infection, there is a probability of 1% that the infection will result in serious compli- cations that requires hospitalization. On average, the hospitals only have capacity to handle 2000 individuals with complications from a cold. Use Little’s law to calculate the average treatment time required to not exceed the capacity.
by the way, I'm not totally comfortable with the wording here. Little's Law deals with averages -- i.e. that the expected number of customers in a queue is equal to the product of the arrival rate and the expected time each customer is in the queue... but the actual thing you are supposed to calculate (which I italicized) really has little to do with averages and technically is worded as a ruin problem. My guess is that its just poorly worded and isn't meant to be a ruin problem but I don't like having to guess these things.

My apologies. So i calculated that the long-run mean fraction of time a single individual has a cold is ##7/107## so i figured ##7/107## or about ##0.065## of the population will be infected in the long term. Does the calculation seem correct?

## What is the birth and death process?

The birth and death process is a mathematical model used to describe the evolution of a system over time. It involves a system that can have a varying number of individuals, with births and deaths occurring randomly over time.

## What is Little's law?

Little's law is a theorem that relates the average number of items in a system, the average time an item spends in the system, and the average rate at which items enter or leave the system. It states that the average number of items in a system is equal to the average time an item spends in the system multiplied by the average rate at which items enter or leave the system.

## How is Little's law used in birth and death processes?

In birth and death processes, Little's law is used to determine the average number of individuals in the system, the average time an individual spends in the system, and the average rate at which individuals enter or leave the system. This information can be used to make predictions about the behavior of the system over time.

## What are some real-world applications of birth and death processes?

Birth and death processes have many real-world applications, such as in population dynamics, epidemiology, and queuing theory. They can also be used to model the behavior of biological systems, such as the growth and decay of cells in a tissue.

## How are birth and death processes different from other mathematical models?

Birth and death processes are different from other mathematical models in that they take into account the randomness of births and deaths over time. This makes them particularly useful for modeling systems where the number of individuals can change over time, such as in populations or queuing systems.

Replies
5
Views
1K
Replies
1
Views
2K
Replies
4
Views
1K
Replies
5
Views
2K
Replies
2
Views
3K
Replies
6
Views
3K
Replies
4
Views
2K
Replies
8
Views
2K
Replies
28
Views
5K
Replies
14
Views
16K