Consider an infinite server queueing system in which customers arrive in accordance with a Poisson process and where the service distribution is exponential with rate $\mu$. Let $X(t)$ denote the number of customers in the system at time $t$. Find (a) $E[X(t+s) \mid X(s)=n]$ (b) $\operatorname{Var}[X(t+s) \mid X(s)=n]$ Hint: Divide the customers in the system at time $t+s$ into two groups, one consisting of "old" customers and the other of "new" customers.
Added by Alex L.
Step 1
The number of "old" customers at time $t+s$ is $n$ times the probability that a customer who was in the system at time $s$ is still in the system at time $t+s$. This probability is $e^{-\mu t}$, because the service times are exponentially distributed with rate Show more…
Show all steps
Your feedback will help us improve your experience
Ameer Said and 54 other Probability educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Recommended Videos
Consider a single server queuing system where customers arrive according to a Poisson process with rate $\lambda$, service times are exponential with rate $\mu$, and customers are served in the order of their arrival. Suppose that a customer arrives and finds $n-1$ others in the system. Let $X$ denote the number in the system at the moment that customer departs. Find the probability mass function of $X$. Hint: Relate this to a negative binomial random variable.
Md.Daniyal A.
Consider a system with Poisson arrivals with a mean equal to 1 customer per minutes and two identical servers with exponential service times equal to 1 minute. The servers can be arranged into two possible arrangements. • Arrivals form a single queue to receive service from the first available server. • The two servers combine to form a single server with a resulting service time that is exponential with a rate equal to 2 customers per minute. Obtain the average number of customers in the system and the average time spent by a customer in the system for each arrangement. Which arrangement will you recommend? Explain.
Lottie A.
Let $X$ and $Y$ be independent exponential random variables with common parameter $\lambda .$ (a) Use convolution to show that $X+Y$ has a gamma distribution, and identify the parameters of that gamma distribution. (b) Use the previous exercise to establish the same result. (c) Generalize part (b): If $X_{1}, \ldots, X_{n}$ are independent exponential rvs with common parameter $\lambda,$ $\quad$ what is the distribution of their sum?
Joint Probability Distributions and Their Applications
Jointly Distributed Random Variables
Recommended Textbooks
Probability with Applications in Engineering, Science, and Technology
Probability and Statistics for Engineers and Scientists
Applied Statistics and Probability for Engineers
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD