Question

In a single-server queuing system, arrivals are Poisson distributed with a mean of 16 per hour and the exponential service time is 3 minutes per person on average. What would be the expected number of persons in the queue (Lq) for the queue disciplines of First-come-first-serve (FCFS) and Last-come-first serve (LCFS)?
1. 4.0, 4.0
2. 4.0, 3.2
3. 3.2, 4.0 
4. 3.2, 3.2

Answer 1

Correct Answer - Option 4 : 3.2, 3.2

Concept:

The number of person in the queue (L_q) is given by

L_q = \(\frac {ρ ^2}{1\ -\ ρ}\)

\(ρ = { λ \over μ }\)

Where, λ = Arrival rate; μ = Service rate; ρ = Utilization of the server;

As this is single server M/M/1 system Lq will be same for FCFS and LCFS.

Calculation:

Given:

λ = 16 person/hr; μ = 1 person per 3 minutes = 20 person/hr;

\(ρ = { λ \over μ } = {16 \over 20}\)

ρ = 0.8

Lq = \(\frac {ρ ^2}{1\ -\ ρ} = \frac {0.8 ^2}{1\ -\ 0.8}\)

Lq = 3.2 

Always convert person per minutes in per hour. Both λ and μ must be in per hour.