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Single Server Poisson Model – 1
(M/M/1) (∞ / FIFO)

P0 = 1 – (λ/μ)   where P0  denotes the probability of system being idle.

Pn = (λ/μ)n P0  = (λ/μ)n {1 – (λ/ μ)}

The quantity λ/μ = ρ is called the traffic intensity.

Average number Ls of customers in the system

Ls    =  E(n) =  n=0  nPn  = λ/(μ- λ)

Average number Lq of customers in the queue

Lq  = E(n) =  n=0  (n-1)Pn  = λ2/{μ (μ- λ)} or  Ls – (λ/μ)

 Average number Lw of customers in the nonempty queue

 Lw  = E{(N-1)/(N-1)>0}  = μ / (μ- λ)

 Probability that the number of customers in the system exceeds k

P(n > k) =  n=k+1  Pn    = ρk+1

 Probability density function of the waiting time in the system:

 f(w) = n=0  f(w/n) Pn =   (μ- λ) e-(μ -λ)w

 Average waiting time Ws of a customer in the system

 Ws  = 1 / (μ- λ)

 Average waiting time Wq of a customer in the queue

 Wq  =  / λ/{μ (μ- λ)}

 Probability that the waiting of a customer in the system exceeds t.

 P(Ws > t) = t   f(w) dw  = e-(μ -λ)t

 Little’s Formula:

Ls   =   λ/(μ- λ)                =  λ Ws

Ls   =  λ/(μ- λ)                = Lq  + λ/μ

Ws =   1 / (μ- λ)             =  Wq + 1/μ

Lq   =   λ2/{μ (μ- λ)}  =  λ Wq

  

Model – 2  (M/M/C) (∞ / FCFS):

 P0 = 1/ [nc-1=0 (1/n!) (λ/μ)n] + [(λ/μ)c (1/c!{1- λ/cμ})]  

 Pn =  (1/c!cn-c) (λ/μ)n P0         If n ≥ c

 Average number Ls of customers in the system :

 Ls    =   [(1/c!c) (λ/μ)c+1 P0 {1/ (1- λ/μc)2}] + λ/μ

 Average number Lq of customers in the queue:

 Lq  =  (1/c!c) (λ/μ)c+1 P0 {1/ (1- λ/μc)2}

 Average Time a customer spends in the system:

 Ws = (1/λ) Ls  =   [(1/μc!c) (λ/μ)c P0 {1/ (1- λ/μc)2}] + 1/μ

 Average Time a customer spends in the Queue:

 Wq = (1/λ) Lq  =   (1/μc!c) (λ/μ)c P0 {1/ (1- λ/μc)2}

 Probability that an arrival has to wait:

P(Ws > 0) =  P(n ≥ c)

                 =  (λ/μ)c P0  / c!(1- λ/μc)

 Probability that an arrival has to get the service without waiting:

 P(getting the service without waiting)  = 1 - P(Arrival has to wait)

              = 1 – {(λ/μ)c P0  / c!(1- λ/μc)}

 Probability that someone will be waiting:

 P(Someone will be waiting )  = P(n ≥ c+1)

                                                = n=c+1  Pn

= (λ/μ)c (λ/μc) P0  / {c!(1- λ/μc)}

 Mean waiting time in the queue for those who actually wait:

 E(Wq /Ws)  =  E(Wq )/P(Ws > 0)

                   = 1/(μc - λ)

 Average number of customers (in non-empty queues), who have to actually wait:

 Lw  = (λ/μc) /(1- λ/μc)

  

 Model – 3

(M/M/1) (N / FIFO)  (Finite capacity, single server Poisson queue model):

 Pn  =  (λ/μ)n  {(1- λ/μ) /(1- (λ/μ)N+1)}

  Pn  =  1/(N+1)  for   λ  =  μ

 Probability that the system is idle:

 P0 = (1 – r) / (1 – (ρ)N+1)  where ρ  = λ/μ

 Average number Ls of customers in the system

Ls    =  E(n) = P0 * nN=0  n ρn

         = [λ/(μ- λ)] – [(N+1) (λ/μ)N+1/(1- (λ/μ)N+1)]  for λ    μ

      = k/2   for   λ  =  μ

 Average queue length:

Lq   =  Ls – λ′/μ . λ′ =  μ(1- P0) the effective arrival rate.

Average waiting time in the system:

Ws  =  Ls / λ′

Average waiting time in the queue:

Wq  =  Lq / λ′

 Average number of units in the system:

  Ls     = [λ/(m- 1)] – [(N+1) (1/m)N+1/(1- (1/m)N+1)] 

Average number of units in the queue:

 Lq   =  Ls – (1 - P0)

  

Model – 4

(M/M/S) (K / FIFO)  or  (M/M/C) (N / FIFO)  :

 P0 = [ns-1=0 (1/n!) (λ/μ)n  +(1/s!) (λ/μ)s   nk=s  (λ/μ)n-s   ]-1

 Pn = (1/n!) (1/m)n  P0                  For  n ≤ s

 Pn = (1/s! sn-s) (1/m)n  P0      For s < n ≤ k

 ρ =  1/ μs

 Lq   =  P0 (1/m)s (r/s!(1 - r )2) [ 1 - ρk – s – (k – s ) (1 – ρ) ρk – s]

 Ls   =   Lq + s [ns-1=0  ( s - n ) Pn]

 Ws  =  Ls / λ′

 λ′  =     μ  [ s - ns-1=0  ( s - n ) Pn]

 Excess capacity or overflow occurs

 P(N= n) =  (1/s! sn-2) (λ/μ)n  P0      

 Non – Markovian Queueing Model 5

(M/G/1): (∞ / GD model)

 Pollaczek-Khinchine formula:

 Ls  =  E(N) = λ E(T) [λ2 {Var(T) + (E(T)2} / 2{1 – λ E(T)}]

 

 









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