Little's Theorem

Little's Theorem (sometimes called Little's Law) is a statement of what was a "folk theorem" in operations research for many years:

N = λT

where N is the random variable for the number of jobs or customers in a system, λ is the arrival rate at which jobs arrive, and T is the random variable for the time a job spends in the system (all of this assuming steady-state). What is remarkable about Little's Theorem is that it applies to any system, regardless of the arrival time process or what the "system" looks like inside. Proof: Define the following:

α ( t ) ≡ number of arrivals in the interval (0,t ) δ ( t ) ≡ number of departures in the interval (0,t ) N ( t ) ≡ number of jobs in the system at time t = α (t ) − δ( t )

γ ( t ) ≡ accumulated customer - seconds in (0,t )

These functions are graphically shown in the following figure: €

The shaded area between the arrival and departure curves is γ (t ) .

λ t = arrival rate over the interval (0,t )

=

α (t ) t

Elec 428

Little’s Theorem

N t = average # of jobs during the interval (0,t ) =

γ (t) t

Tt = average time a job spends in the system in (0,t )

€

=

γ (t) α (t)

€

⇒ γ ( t ) = Ttα ( t ) T α (t ) ⇒ Nt = t = λt Tt t

Assume that the following limits exist:

€ lim λt = λ

t →∞

lim Tt = T

t →∞

Then

€

lim N t = N

t →∞

also exists and is given by N = λT .

€

Keywords: Little's Law Little's Theorem Steady state

Page 2 of 2

Little's Theorem (sometimes called Little's Law) is a statement of what was a "folk theorem" in operations research for many years:

N = λT

where N is the random variable for the number of jobs or customers in a system, λ is the arrival rate at which jobs arrive, and T is the random variable for the time a job spends in the system (all of this assuming steady-state). What is remarkable about Little's Theorem is that it applies to any system, regardless of the arrival time process or what the "system" looks like inside. Proof: Define the following:

α ( t ) ≡ number of arrivals in the interval (0,t ) δ ( t ) ≡ number of departures in the interval (0,t ) N ( t ) ≡ number of jobs in the system at time t = α (t ) − δ( t )

γ ( t ) ≡ accumulated customer - seconds in (0,t )

These functions are graphically shown in the following figure: €

The shaded area between the arrival and departure curves is γ (t ) .

λ t = arrival rate over the interval (0,t )

=

α (t ) t

Elec 428

Little’s Theorem

N t = average # of jobs during the interval (0,t ) =

γ (t) t

Tt = average time a job spends in the system in (0,t )

€

=

γ (t) α (t)

€

⇒ γ ( t ) = Ttα ( t ) T α (t ) ⇒ Nt = t = λt Tt t

Assume that the following limits exist:

€ lim λt = λ

t →∞

lim Tt = T

t →∞

Then

€

lim N t = N

t →∞

also exists and is given by N = λT .

€

Keywords: Little's Law Little's Theorem Steady state

Page 2 of 2

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