Class Notes � Server Utilization

Notation

A/B/c/N/k

Let�s put it together!

If �������������������������������������������� , then everything�s fine.

If �������������������������������������������� , then more arrivals than can be serviced.

Let p_n: probability that there�s n in the system.� (That�s �p� sub zero!)

Let

p₀: probability that there�s zero in the system.

�

p_n: probability that there�s n in the system.

What is the relationship between p₀ and p₁?

l p₀ = p₁ m

or

l p₀ = m p₁

and

(l + m) p_n = l p_n-1 + m p_n+1, where n = 1, 2, 3, �

l p₀ = m p₁

����������������������������������� equations or ���������������� ����������� equations or ���������������� �equations.

Rewrite:

l p₀ = m p₁

p₁ = ����������������������������

Now introducing rho (r) � the server utilization:

r = l/m, for 1 server!

r + 1 = l/m + m/m = (l+ m)/m

r + n = l/m + nm/m = (l+ nm)/m

So,

p₁ = (r) p₀

In this equation,

(l + m) p_n = l p_n-1 + m p_n+1, for n = 1

(l + m) p₁ = l p₀ + m p₂

p₂ = [(l + m)/m] p₁ � (l/m) p₀ = [r + 1] p₁ � (r) p₀ = [r + 1] (r) p₀ � (r) p₀

=� r² p₀ + rp₀ � rp₀ = r² p₀

In general,

p_n = ���������������������������������������� (in terms of r)

For the summation in the range 0 <= i < �:

S p_i = 1

S rⁱ p₀ = 1

p₀ S rⁱ = 1

The term S rⁱ� is an ���������������������������������������������������� �and becomes smaller as i becomes larger.

Remember:

Sum = S = (1^st term)/(1 � r), where |r| < 1

So:

S rⁱ� = 1/(1 - r), for |r| < 1

Then, for above:

p₀ S rⁱ = 1

p₀ [1/(1 - r)] = 1, for |r| < 1

p₀ [1/(1 - r)] = 1

p₀ /(1 - r) = 1

p₀ = 1 - r

So, in general:

p_n = ���������������������������������������������������� , for n = 0, 1, 2, �, �, and r < 1

p_n is now in terms of server utilization!

For parallel servers, when is the system stable?

r is stable when it is less than 1.

So, stable when:

r < 1

l/(cm) < 1

l < cm

or

c > l/m

(# servers) (is greater than) (traffic intensity)

Ex.� Let l = 3, m = 4, and c = 3.� Is the system stable?

r is stable when it is less than 1.

So, stable when:

r < 1

r = l/(cm) = 3/(3�4) = 0.75

Stable!

What about for l = 5, m = 4, and c = 3.� When is the system stable?