Queueing Theory. Frank Y. S. Lin Information Management Dept. National Taiwan University [email protected] References. Leonard Kleinrock, “ Queueing Systems Volume I: Theory ”, New York: Wiley, 1975-1976.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Volume I: Theory”, New York: Wiley, 1975-1976.
Network Delay =
+ Propagation Delay (depends on the distance)
+ Node Delay Processing Delay
(independent of packet length,
e.g. header CRC check)
Adapter Delay (constant)
FX(x; t + τ) = FX(x; t)
FX(x; t) = FX1,…, Xn(x1,…, xn ; t1,…,tn)
= FX1(x1; t1) …FXn(xn; tn)
P[X(tn+1) = xn+1 | X(tn) = xn, …., X(t1) = x1]
= P[X(tn+1) = xn+1 | X(tn) = xn]
If X ~Exp(λ), for any a,b > 0,
P[X > a + b | X > a] = P[X > b]
P[X > a + b | X > a]
rate out of state j = rate into state j
rate in = rate out
System time T
Single Server, Single Queue
(The Classical Queueing System)
rate inM/M/1 Queue (cont’d)
(n-1in the queue and 1 in service)
S0 = empty system
(rate in = rate out)
Single Server, Finite Storage
λk = λ if k < K
0 if kK
μk = μ
M/M/∞ - Infinite Servers, Single Queue
M/M/m - Multiple Servers, Single Queue
(The m-Server Case)
λk = λ
μk = kμ if km
mμ if k > m
For k > m
Total system time =
If we have 4 M/M/1 systems: 4 parallel communication links that can each handle 50 pps (μ), arrival rate λ = 25 pps per queue.
average delay = 40 ms.
Whereas for an M/M/4 system,
average delay = 21.7 ms.
If we have an M/M/4 system with service rate μ=50 pps for each server, and another M/M/1 system with service rate 4μ = 200 pps. Both arrival rate is λ = 100 pps
delay for M/M/4 = 21.7 ms
delay for M/M/1 = 10 ms
If we have nM/M/1 system with service rate μ pps for each server, and another M/M/1 system with service rate nμ pps. Both arrival rate is nλ pps
Multiple Servers, No Storage
(m-Server Loss Systems)