Queuing Theory Equations Definition = Arrival Rate = Service Rate - - PDF document

Queuing Theory Equations Definition λ = Arrival Rate μ = Service Rate ρ = λ / μ

C = Number of Service Channels M = Random Arrival/Service rate (Poisson) D = Deterministic Service Rate (Constant rate) M/D/1 case (random Arrival, Deterministic service, and one service channel) Expected average queue length E(m)= (2ρ- ρ2)/ 2 (1- ρ) Expected average total time E(v) = 2- ρ / 2 μ (1- ρ) Expected average waiting time E(w) = ρ / 2 μ (1- ρ) M/M/1 case (Random Arrival, Random Service, and one service channel) The probability of having zero vehicles in the systems Po = 1 - ρ The probability of having n vehicles in the systems Pn = ρn Po Expected average queue length E(m)= ρ / (1- ρ) Expected average total time E(v) = ρ / λ (1- ρ) Expected average waiting time E(w) = E(v) – 1/μ

M/M/C case (Random Arrival, Random Service, and C service channel) Note : c ρ must be < 1.0 The probability of having zero vehicles in the systems Po =

( )

1 _ 1

/ 1 ! ! ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − +

∑

− = c n C n

c c n ρ ρ ρ The probability of having n vehicles in the systems Pn = Po ! n

n

ρ for n < c Pn =Po ! c c

c n n −

ρ for n > c Expected average queue length E(m)=

( )

2 1

/ 1 1 ! c cc P

c

• ρ

ρ −

+

Expected average number in the systems E(n) = E(m) + ρ Expected average total time E(v) = E(n) / λ Expected average waiting time E(w) = E(v) – 1/μ

M/M/C/K case (Random Arrival, Random Service, and C service Channels and K maximum number of vehicles in the system) The probability of having zero vehicles in the systems For 1 ≠ c ρ

1 1 1

1 1 ! ! 1

− − = + −

⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∑

c n c K c n

• c

c c n P ρ ρ ρ ρ For 1 = c ρ

( )

1 1

1 ! ! 1

− − =

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∑

c n c n

• c

K c n P ρ ρ c n for ! 1 ≤ ≤ =

• n

n

P n P ρ k n c for P ! c 1

• n

c

• n

≤ ≤ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ρ c P

n

( )

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =

− + − c k c k c

• c

c k c c c c c P m E ρ ρ ρ ρ ρ ρ 1 1 1 1 ! ) (

1 2

∑

− =

− − + =

1

! ) ( ) ( ) (

c n n

• n

n c P c m E n E ρ

( )

K

P n E v E − = 1 ) ( ) ( λ μ 1 ) ( ) ( − = v E w E