LECTURE  No. 28

Held on: Monday, November 13, 2000.

Prepared by : Robin Varghese& Amit Bhatnagar

 

 

In a queueing system:

 

T_i : Time in state i

Let N_n-1,n = Number of transitions from n-1 to nth state

N_n-1,n + N_n+1,n – N_n     £ 1             { transitions into a state –Transitions out }

Taking expectations,

 

 E (N_{n-1, n}) + E (N_{n+1, n}) – E (N_n)      £ 1

  T                              T

 

As T       infinity

 E (N_n-1) _n) + E (N_{n+1, n})                           E( N_n)                       (1)

  T                             T

 

 

T_n-1 = is time spent in state n-1

Then E N_n-1,n  = l_n-1 T_n-1  = Average number of people who arrive in  time when state is “n-1”.

 

Therefore (1) becomes,  l_n-1 T_n-1 + m_n+1 T_n+1  =        (l_n + m_n) T_n

                                        T                                T

 

T_n         P_n   Probability of being in state n.

T                              

 

l_n-1 P_n-1 + m_n+1 P_n+1  = (l_n + m_n) P_n

 

This should be true for all n.

Hence, l_o Po = m₁ P₁

 

Solving this we get,        P_I­ = l_o l₁ …. l_i-1         P₀

                                                                                m_{1…………..}m_i

                        

                                             

 

Po =                      1

                        µ

                    1+S    l_o ……. l­_i-1

                      i=1  m_1………..m_i

 

steady  - State measures of performance

 

(i)       Expected waiting time in system = W_s

(ii)             

^ m

W_q = Expected waiting time in Queue

{W_s = W_q +     }

(iii)            L_s = Expected no. of customers in system

(iv)             L_q = Expected no. of customers in queue

 

{ L_s = 1 + L_q} = not true when no customers in queue

 

       

 

 

         µ

N=1

:. L_s = S n P_n (Probability of finding ‘n’ people in the system  times ‘n’).

	µ
			µ

 

 

n-1

L_q = S       (n-c) p_n = S   n p_n+c

 

 

n=c+1

Little’s Law

 

	µ

 

n=o

L_s = l_eff W_s        ,where   l_eff   = S l_nP_n (Effective arrival rate)

L_q = l_eff W_q

 

 

                                   t

 

Avg. queue length =  1   ∫ Q (T) dT     ,i.e., (Area)_t

0

                                  t                                  t

	N(t)		N(t)

 

                                               

i=1

    (Area)_t  

i=1

=  S W_i                       =          S    W_i       *   N(t)                                    - (1)

          t             t                                N(t)           t

                

Where N(t) = no. of arrivals in time t

:. Eq. (1) becomes

(Area)_t

     t            = W_s    l_eff

  

 

  L_s = W_s  l_eff

 

:. L_s – L_q = leff = (expected no. of busy servers)

                  m  

M/M/1 (GD/µ/µ)

l_n = l

m_n = m

                                              l

Define traffic intensity ρ =      m

 

                                         

 

Po

                                 l_o ……l_n-1              

P_n = PⁿP_o     { P_n =      m₁ …… m_n                    }

 

                              

n=0

µ

      Po   S  Pⁿ = 1   Po = 0         P_n = 0

 

Po = (1-P)

P_n = (1-ρ) ρ ⁿ                               l_eff = l

µ

n=1

L_s = S  n (1 – ρ) ρ ⁿ = ρ         Lq = L_s – ρ =  ρ ²

                             (1- ρ)                         (1- ρ)

 

 

:.W_s =    1                       :. W_q =   ρ

          m (1- ρ)                            m (1- ρ)

 

 

Find p.d.f. of activity time for FCFS:

 

X_I in exp. (m)

 

Ù

Then,

 

S       

i-1

X_I­ ~ F long dist (n, m)

 

 

F ( n, m)

 

p.d.f. is

 

          f (t) = m (mt)ⁿ e­^-mt

                                        n!

 

 

 

Show. That p.d.f. X₁ + X₂ is m² t e ^-mt

                                                                          2

 

find p.d.f. of waiting time 

when  P (W @ X)

Profitability of finding 5 people in system : p⁵ (1-p)

µ

{ “ PASTA” = Poisson arrivals see time average}

n=0

P (w@x) = S     p (w@x, he sees n people in system on arrival)

 

Some important Points:

 

·         FCFS minimizes large waiting times.

·         LCFS minimizes mean waiting time.