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)= (2 ρ - ρ 2 )/ 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 P o = 1 - ρ The probability of having n vehicles in the systems P n = ρ n P o 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 must be < 1.0 Note : c The probability of having zero vehicles in the systems ⎡ ⎤ _ 1 ρ ρ − n C c 1 ∑ + ⎢ ⎥ P o = ( ) − ρ ⎣ ⎦ n ! c ! 1 / c = n 0 The probability of having n vehicles in the systems ρ n P n = P o for n < c n ! ρ n P n =P o for n > c − n c c c ! Expected average queue length + ρ c 1 1 P E(m)= ( ) o − ρ 2 cc ! 1 / c 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 − 1 ⎡ ⎤ ⎛ ⎞ − + ρ K c 1 ⎛ ⎞ ⎜ ⎟ − ⎢ ⎜ ⎟ ⎥ 1 ρ ⎛ ⎞ − ⎛ ⎞ ρ ⎜ ⎟ = ∑ c 1 c ⎝ ⎠ ⎢ 1 ⎥ c ≠ ρ + ⎜ ⎟ ⎜ ⎟ n For 1 P ⎜ ⎟ ⎜ ⎟ ⎢ ρ ⎥ o ⎝ ⎠ ⎝ ⎠ c n ! c ! − ⎜ ⎟ = ⎢ n 0 1 ⎥ ⎜ ⎟ c ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ − 1 ⎡ ⎛ ⎤ ρ ρ ⎞ − ⎛ ⎞ = ∑ c c 1 1 ( ) ⎜ ⎟ = ρ + − + ⎜ ⎟ ⎢ n ⎥ For 1 P ⎜ ⎟ K c 1 o ⎝ ⎠ ⎝ ⎠ c ⎣ n ! c ! ⎦ = n 0 1 = ρ ≤ ≤ n P P for 0 n c n o n ! ⎛ ⎞ 1 ρ ≤ ≤ = ⎜ ⎟ n P P for c n k n ⎝ ⎠ o n - c c c ! ρ ⎛ ⎞ ρ ⎜ ⎟ c P ⎡ − + − ⎤ ρ k c 1 ρ ρ k c ⎛ ⎞ ⎛ − ⎞ ⎛ ⎞ o ⎝ ⎠ ( ) c = − − − + ⎢ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎥ E ( m ) 1 1 k c 1 ρ 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ − ⎞ ⎢ ⎥ c c c ⎣ ⎦ ⎜ ⎟ c ! 1 ⎝ ⎠ c − − ρ n c 1 ∑ ( c n ) = + − E ( n ) E ( m ) c P o n ! = n 0 E ( n ) = E ( v ) ( ) λ − 1 P K 1 = − E ( w ) E ( v ) μ