Pure Birth Process Lecture Notes PDF

# Pure Birth Process - **LECTURE IV** - **Lecturer** - Prof. A-B. Alhassan / Mr. S. Ibrahim ## Table of Contents 1. Introduction to Pure Birth Process 2. Pure Birth Process Overview 3. Mathematical Formulation 4. Rule (i): Probability of One Addition 5. Rule (ii): Probability of More than One Addit...

# Pure Birth Process - **LECTURE IV** - **Lecturer** - Prof. A-B. Alhassan / Mr. S. Ibrahim ## Table of Contents 1. Introduction to Pure Birth Process 2. Pure Birth Process Overview 3. Mathematical Formulation 4. Rule (i): Probability of One Addition 5. Rule (ii): Probability of More than One Addition 6. Generalized Poisson Process 7. Example Setup: Simulating Population Growth 8. Discrete Time Simulation Example 9. Results from the Simulation 10. Probabilistic Calculations for One Addition 11. Example: Probability of Multiple Additions 12. Comparing Pure Birth vs Poisson Process ## Introduction to Pure Birth Process - **Definition** - A special type of stochastic process used to model certain types of systems, particularly in the context of queues or population dynamics. - In the realm of queues and probability theory, it is a specific kind of Markov process that describes how a population (or the number of customers in a system) increases over time, with the assumption that no customers ever leave the system and they can only arrive. - **Key Features of the Pure Birth Process** - **No departures:** In a pure birth process, individuals or customers can only enter the system (arrivals), and there are no departures. - **Discrete states:** The system typically consists of a sequence of states representing the number of entities (or customers) in the system at any given time. - **Rate of arrivals:** The rate at which entities arrive in the system depends on the state. Typically, the transition rates from state *n* to state *n+1*. - **Markov property:** The process satisfies the Markov property, meaning the future state of the process only depends on the current state and not on the sequence of events that preceded it. ## Mathematical Representation - **Where** - *X(t)* be the number of entities in the system at time *t*. - The system transitions from state *n* to state *n+1* at rate λ_*n*. - The transition rate from state *n* to state *n+1* is: - *P(X(t+Δt)=n+1|X(t)=n)=Δt+o(Δt)*_*n* - The process evolves over time, and the total rate of arrivals into the system can be described by a set of rates λ₀,λ₁,λ₂,..., where λ_*n* is the rate at which the system moves from state *n* to state *n+1*. - **Mathematical Formulation** - **Population Size:** Denoted by *M(t)*, where *t* is time and *M(t)* is the number of individuals at time *t*. - **Probability of Addition:** The probability that one individual is added is *P₁(t)*, which depends on time. - **Probability of Multiple Additions:** The probability that more than one individual is added at time *t* is *P_k(t)*, where *k>*1. - **Assumption:** The additions follow a Poisson-like process, but with modified rules. ## Applications of Pure Birth Process - **Population growth:** In biological contexts, a pure birth process can model the growth of a population where new individuals are born but no individuals die. - **Queueing theory:** It can be used to model systems where customers only arrive (e.g., an "infinite server" queue model where once a customer enters the system, they don't leave). - **Reliability models:** It can describe the process of adding new components to a system that has no failure (i.e., components only arrive and never leave). ## Related Processes - **Birth-Death Process:** The pure birth process is a special case of the birth-death process, where there are both births (arrivals) and deaths (departures). - **Poisson Process:** If the birth rates λ_*n* are constant, then the pure birth process becomes a Poisson process, which models random arrivals at a constant rate. ## Rule (i): Probability of One Addition - **Probabilistic Model** - *P₁(t)* is the probability that a single addition happens at time *t*. - *P₁(t)* depends on the current population size and the elapsed time. - Example form: *P₁(t)=λ·f(t)*, where λ is a rate constant and *f(t)* is a time-dependent function. ## Rule (ii): Probability of More than One Addition - **Multiple Additions** - The probability of more than one individual being added at time *t* is described by *P_k(t)* for *k>*1. - *P_k(t)* generally has the form: - *P_k(t) = (λ· f(t))^k / k! * e^{-λ· f(t)} - This follows a generalized Poisson distribution. - **Example: Hash Collision in a Hash Table** - Let's say we have a hash table of size 10 and a hashing function that maps integers to indices within this table. When we insert elements into the hash table, the "addition" refers to inserting an item at a specific index. - In this case, Rule (ii) would apply if we want to calculate the probability of a hash collision that is, the probability that more than one item hashes to the same index (i.e., more than one addition happens at the same position). - **Scenario:** - We are inserting 3 items into the hash table. - The size of the hash table is 10, so there are 10 possible hash indices (0 to 9). - We want to calculate the probability that at least two of the three inserted items hash to the same index (i.e., a collision occurs). - **Solution** - Each of the 3 items can be hashed to any of the 10 indices, so the total number of ways to assign hash values (without any restrictions) is: 10×10×10=1000 - items with no collision is: 10×9×8=720 - The probability of no collision occurring is the ratio of the favorable outcomes (no collision) to the total possible outcomes: *P(no collision) = 720/1000 = 0.72* - The probability of more than one addition (i.e., at least one collision) is the complement of the probability of no collision: *P(collision)=1-P(no collision)=1-0.72=0.28* ## Generalized Poisson Process - **Connection to Poisson Process** - The Pure Birth Process generalizes the Poisson process by allowing time-dependent probabilities. - In a standard Poisson process, events (births) occur independently with a constant rate. - In a Pure Birth Process, the rate varies with time and population size. ## Discrete Time Simulation Example - **Time Steps:** Simulate population growth in discrete time intervals. - **Simulation Process:** - At each time step, calculate *P₁(t)* and *P_r(t)* - If *P₁(t)* is greater than a random value *r* (uniformly distributed), add one individual to the population. - Repeat for each time step. - **Results from the Simulation** - The population increases over time. - The growth rate is influenced by both the birth rate constant *λ* and the time-dependent factor *f(t)*. - Plot the results to show exponential like growth with some variation due to the time varying birth rate. ## Example Setup: Simulating Population Growth - **Scenario** - A computer simulation modeling a growing population of bacteria. - *λ=0.1* (birth rate constant). - Time-dependent function: *f(t)=e^0.02t* (reflecting environmental factors that change the birth rate over time). - Population growth rate is: - *dP/dt = P(t)·f(t)* - *dP/dt = 0.1P(t)·e^0.02t* - To solve this differential equation, we can separate the variables *P(t)* and *t*: - *dP/P(t) = 0.1e^0.02tdt* - Integrate both sides: *ln|P(t)| = 5e^0.02t + C*, where *A=e^C* - *P(t) = e^5e0.02t + c* - *P(t) = Ae^5e0.02t* ## Probabilistic Calculations for One Addition - **What is Theoretical measurement** - At time *t=50*, calculate the probability of one addition. - Given: - *λ=0.1* - *(50)=0.02t* - *P₁(50)=0.1·e^0.02×50≈0.1·e¹≈0.1x2.718=0.2718* - **Example: Probability of Multiple Additions** - Calculate the probability of more than one addition. - For *k=2* - *P₂(50) = (0.1·e¹)²/2! · e^-0.1·e¹* - ≈0.0368 ## Comparing Pure Birth vs Poisson Process - **Step by Step** - **Poisson Process:** Constant rate of addition (no time dependency). - **Pure Birth Process:** Rate · *f(t)*, where *f(t)* can vary with time. - The Pure Birth Process generalizes the Poisson process by introducing time-varying rates. ## Thank You! - **Questions Time**