Performance analysis: Average queue length and service time

Reason to study queueing theory
- Purpose of queueing models:

Commonly used performance measures

Common performance measures:

Average queue length N

N = mean (expected) number of customer = ∑_{{k = 0, 1, .., ∞}} k × Ҏ[ k customers in system] (definition of "expected value") = ∑_{{k = 0, 1, .., ∞}} k × p_k

This is the average number of customers in the system.

Average service time T

T = average time that a customer spends in the system N = --- (Little's formula) λ

This is the average amount of time that a customer spends in the system.

Example: Average queue length in the M/M/1 queue

Recall that the state probability i of the M/M/1 queue is

Ҏ[ k customers in system] = ρ^k (1 - ρ) (see: click here)

The average (expected) queue length N of the M/M/1 queue is:

N = ∑_{{k = 0, 1, .., ∞}} k × p_k (definition of N) = ∑_{{k = 0, 1, .., ∞}} k × ρ^k (1 - ρ) = (1 - ρ) × ∑_{{k = 0, 1, .., ∞}} k × ρ^k .... (1)

U = ∑_{{k = 0, 1, .., ∞}} ρ^k = 1/(1 - ρ) (see: click here) Or: U = ∑_{{k = 0, 1, .., ∞}} ρ^k = (1 - ρ)^-1
Thus: dU ---- = ∑_{{k = 0, 1, .., ∞}} k ρ^k-1 = (1 - ρ)^-2 dρ Therefore: ∑_{{k = 0, 1, .., ∞}} k ρ^k = ρ × (1 - ρ)^-2

Substitute in (1): N = (1 - ρ) × ∑_{{k = 0, 1, .., ∞}} k × ρ^k ..... (1) = (1 - ρ) × ρ × (1 - ρ)^-2 = ρ × (1 - ρ)^-1
Summary: mean number of customers in an M/M/1 queue

ρ N(M/M/1) = ------- ..... (2) 1 - ρ

Example: Average service time of a customer in the M/M/1 queue

According to the Little's formula:

N = λ T (Little's formula)

the average service time of the M/M/1 queue is

N T = --- λ 1 ρ = --- * ------- λ 1 - ρ 1 λ/μ = --- * ------- λ 1 - ρ 1 1 = --- * ------- ..... (3) μ 1 - ρ

Derived performance measures

Idle system:

The system is idle when there are zero (0) customers in the system

The fraction of time that the system is idle is equal to the probability that the system is empty

Therefore:

fraction of time that system is idle = Ҏ[ 0 custumer in system ] = p₀

The fraction of time that an M/M/1 queue is idle is equal to:

fraction of time that M/M/1 queue is idle = 1 - ρ

Busy system: (busy = not idle)

The system is idle when there are not zero (0) customers in the system

The fraction of time that the system is idle is equal to the probability that the system is not empty

Therefore:

fraction of time that system is idle = 1 - Ҏ[ 0 custumer in system ] = 1 - p₀

The fraction of time that an M/M/1 queue is busy is equal to:

fraction of time that M/M/1 queue is idle = ρ