PDF Queuing Theory Equations - DIMACS

[Pages:3]Queuing Theory Equations Definition = Arrival Rate = Service Rate = /

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- 2)/ 2 (1- ) Expected average total time E(v) = 2- / 2 (1- ) Expected average waiting time E(w) = / 2 (1- )

M/M/1 case (Random Arrival, Random Service, and one service channel)

The probability of having zero vehicles in the systems Po = 1 - The probability of having n vehicles in the systems Pn = n Po Expected average queue length E(m)= / (1- ) Expected average total time E(v) = / (1- ) Expected average waiting time E(w) = E(v) ? 1/

M/M/C case (Random Arrival, Random Service, and C service channel)

Note : must be < 1.0 c

The probability of having zero vehicles in the systems

Po =

c-1 n=0

n n!

+

C

_1

c!(1- / c)

The probability of having n vehicles in the systems n

Pn = Po n! n

Pn =Po c n - c c !

for n < c for n > c

Expected average queue length

c+1

1

E(m)= Po cc! (1 - / c)2

Expected average number in the systems

E(n) = E(m) +

Expected average total time E(v) = E(n) /

Expected average waiting time E(w) = E(v) ? 1/

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

Po

=

c

-1

n=0

1 n!

n

+

c c!

1

-

c 1-

K -c+1

c

-1

For = 1 c

Po

=

c-1

n=0

1 n n!

+

c c!

(K

-c

+ 1) -1

Pn

=

1 n!

n

Po

for 0 n c

Pn

=

c

1

n-c

c!

n Po

for c n k

E(m)

=

Po

c

c

c!1- 2

1

-

c

k -c+1

- 1-

c

(k

-c

+ 1)

c

k

-c

c

E(n)

=

E(m)

+

c

-

Po

c -1 n=0

(c

- n) n!

n

E(v)

=

E(n)

(1- PK

)

E(w) = E(v) - 1

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