Queuing Theory Equations - DIMACS

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- ρ )/ 2 (1- ρ) 2

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 = ρ Po n

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)

ρ

Note :

must be < 1.0

c

The probability of having zero vehicles in the systems

⎡ c −1 ρ n ⎤ ρC + Po = ⎢∑ ⎥ ⎣ n = 0 n! c!(1 − ρ / c ) ⎦

_1

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

Pn =Po

ρn

for n < c

n!

ρ c

n

n − c

for n > c

c!

Expected average queue length E(m)= Po

ρ c +1

1

cc! (1 − ρ / c )2

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

For

Pn =

ρ c

ρ c

≠1

⎡ ⎛ ⎛ ρ ⎞ K −c +1 ⎞⎤ ⎜1− ⎜ ⎟ ⎟⎥ ⎢ c −1 c ⎞ ⎛ ρ 1 c ⎜ ⎟⎥ ⎛ ⎞ Po = ⎢∑ ⎜ ρ n ⎟ + ⎜⎜ ⎟⎟⎜ ⎝ ⎠ ⎟⎥ ⎢ n =0 ⎝ n! ⎠ ⎝ c! ⎠ ρ − 1 ⎜ ⎟⎥ ⎢ ⎜ ⎟ c ⎢⎣ ⎝ ⎠⎥⎦

=1

⎡ c −1 ⎛ 1 ⎞ ⎛ ρ c ⎞ ⎤ Po = ⎢∑ ⎜ ρ n ⎟ + ⎜⎜ ⎟⎟(K − c + 1)⎥ ⎣ n =0 ⎝ n! ⎠ ⎝ c! ⎠ ⎦

1 n ρ Po n!

−1

−1

for 0 ≤ n ≤ c

⎛ 1 ⎞ Pn = ⎜ n -c ⎟ ρ n Po ⎝ c c! ⎠

for c ≤ n ≤ k

⎛ρ⎞ Po ρ c ⎜ ⎟ ⎡ k − c +1 k −c c⎠ ⎛ρ⎞ ⎤ ⎛ ρ⎞ ⎛ρ⎞ ⎝ 1− ⎜ ⎟ E ( m) = − ⎜1 − ⎟(k − c + 1)⎜ ⎟ ⎥ 2 ⎢ ⎝ c ⎠ ⎦⎥ ⎝ c⎠ ⎛ ρ ⎞ ⎣⎢ ⎝ c ⎠ c!⎜1 − ⎟ ⎝ c⎠

(c − n ) ρ n n! n =0 c −1

E (n) = E (m) + c − Po ∑ E (v ) =

E ( n) λ (1 − PK )

E ( w) = E (v) −

1

μ