Queueing System Analysis: Average Customers Calculation
This solution explains how to calculate the average number of customers in a queueing system using the provided information about the queue length, arrival rate, and departure rate.
Key Formula
The average number of customers in the system (L) is the sum of the average number of customers waiting in the queue (Lq) and the average number of customers currently being served. In many standard queueing models, the average number of customers being served can be represented by the ratio of the arrival rate (λ) to the departure rate (μ) when the system is in a steady state. The formula is:
$L = L_{q} + \frac{\lambda}{\mu}$
Given Information
- Average customers in the queue, Lq = 30
- Arrival rate of customers, λ = 16 customers per hour
- Departure rate of customers, μ = 32 customers per hour
Step-by-Step Calculation
- Identify the variables: We have Lq = 30, λ = 16, and μ = 32.
- Calculate the average number of customers being served: This is given by λ/μ.
$\frac{\lambda}{\mu} = \frac{16}{32} = 0.5$
- Calculate the average number of customers in the system (L): Add the average number in the queue to the average number being served.
$L = L_{q} + \frac{\lambda}{\mu} = 30 + 0.5 = 30.5$
Result
The average number of customers in the system is 30.5.
Important Questions from Queueing Theory
The probability of getting a total of 7 on two dice thrown together is:
If moment generating function of continuous random variable X is \(\frac{λ}{λ-t}\) t < λ, then E(X 3) equals to:
If moment generating function of discrete random variable X is (q + pe t) n, then E(X 2) equal to
If A and B are mutually exclusive events such that P(A) P(B) > 0, then which option is correct?
Two random variables X and Y are said to be independent if:
