Statistics II · Unit 6 · 7 hrs
Stochastic Process
Exam-focused notes for Stochastic Process (Statistics II, STA215): what the TU syllabus asks and how it has actually been tested, with 9 solved past questions from this unit.
What this unit covers
- Definition and classification
- Markov Process: Markov chain, Matrix approach, Steady-State distribution
- Counting process: Binomial process, Poisson process
- Simulation of stochastic process
- Queuing system: Main component of queuing system, Little's law
- Bernoulli single server queuing process: system with limited capacity
- M/M/1 system: Evaluating the system performance
M/M/1 system
Explain briefly the queuing theory. Customers arrive at a one-man barber shop according to Poisson process with mean inter arrival time of 12 minutes. Customer spends an average of 10 minutes in the barber’s chair. What is the expected number of customers in the barber shop in the queue? [5]
Queuing theory is the mathematical study of waiting lines. It models systems in which customers (or jobs) arrive, possibly wait in a queue, receive service from one or more servers, and then leave. Its aim is to predict system performance measures such as a...
Full solved answer →What are the basic concepts of queuing theory? In a super market, the average arrivals rate of customer is 10 per every 30 minutes following Poisson process. The average time taken by the cashier to list and calculate the customers purchase is 2.5 minutes following exponential distribution. What is the probability that queue length exceeds 6? What is the expected time spent by customer in the system? [5]
- Arrival: 10 customers per 30 minutes (Poisson process) - Service time: 2.5 minutes per customer (exponential distribution) - Find: $P(\text{queue length} 6)$ and expected time in system $W$ --- Queuing theory is the mathematical study of waiting lines. A ...
Full solved answer →Counting process
Customers of certain Internet service provider connect to the internet at the average rate of 10 new connections per minute. Connections are modelle by binomial counting process. a. What frame length gives the probability 0.1 of an arrival during given frame? b. Find the mean and variance for the number of seconds between two consecutive connections. [5]
- Arrival rate: $\lambdaA = 10$ connections per minute - Process model: Binomial counting process - Part (a): required frame probability $p = 0.1$ - Part (b): find mean and variance of inter-arrival time in seconds Convert rate to seconds for consistency: $...
Full solved answer →Markov Process
Define Markov chain and its characteristics. [5]
A stochastic process is a family of random variables indexed by time as a parameter. It is denoted as {X(t), t ∈ T}, where: - T = index set (time parameter) - X(t) = random variable at time t - I = state space (the set of all possible values assumed by X(t)...
Full solved answer →Define the Markov chain and introduce its basic notations. Also, explain the characteristics of a Markov chain. [5]
A Markov chain is a stochastic process {X(t), t = 0, 1, 2, ...} that satisfies the Markov property (memoryless property), which states that the future state of the process depends only on the present state and not on the past states. Mathematically, for all...
Full solved answer →Define Markov chain and describe its characteristics. [5]
A Markov Chain is a special type of stochastic process in which the future state of the system depends only on the present state, and not on the past states (history). It is a discrete-time stochastic process {X(t), t = 0, 1, 2, ...} that satisfies the Mark...
Full solved answer →Every day is generally considered as either sunny or rainy. A sunny day is followed by another sunny day with probability 0.8 whereas a rainy day is followed by a sunny day with probability 0.4. Suppose it rains on Monday. Make forecasts for Tuesday and Wednesday. [5]
Given data: - States: Sunny (S), Rainy (R) - P(Sunny Sunny) = 0.8, therefore P(Rainy Sunny) = 0.2 - P(Sunny Rainy) = 0.4, therefore P(Rainy Rainy) = 0.6 - Initial condition: it rains on Monday, so P(Rainy on Monday) = 1 - Required: forecast for Tuesday and ...
Full solved answer →Queuing system
Define main component of queuing system. [5]
A queuing system is a system designed to perform certain tasks or process certain jobs by one or several servers, where jobs wait in a queue to be processed. The main components of a queuing system are described below: --- - This component describes how cus...
Full solved answer →Bernoulli single server queuing process
Jobs are sent to mainframe computer at a rate of 4 jobs per minute. Arrivals are modeled by a binomial process. a. Choose a frame size that makes the probability of a new received during each frame equal to 0.1. b. Using the chosen frame compute the probability of more than 4 jobs received during one minute. c. Compute mean and variance of inter arrival time? [5]
- Arrival rate: $\lambda = 4$ jobs per minute - Process type: Binomial (Bernoulli frames) - Target frame probability: $p = 0.1$ --- In a binomial process, arrivals occur one per frame with probability $p$, so the rate is: $$\lambda = \frac{p}{\Delta} \impli...
Full solved answer →Make Unit 6 stick
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