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
[continues]
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
[continues]
Cite This Essay
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(2012, 12). Effect of Mobile Phones on Life. StudyMode.com. Retrieved 12, 2012, from http://www.studymode.com/essays/Effect-Of-Mobile-Phones-On-Life-1273675.html
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"Effect of Mobile Phones on Life." StudyMode.com. 12, 2012. Accessed 12, 2012. http://www.studymode.com/essays/Effect-Of-Mobile-Phones-On-Life-1273675.html.
