Page 1 of 2

Effect of Mobile Phones on Life

Continues for 1 more pages »
Read full document

Effect of Mobile Phones on Life

Page 1 of 2
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

Rate this document

What do you think about the quality of this document?

Share this document

Let your classmates know about this document and more at Studymode.com