1. Handling Routing Transport Haifa JFK TLV BGN To: Yishay From: Vered
Shipment 792
Pack. 1 of 3
Shipment 792
Pack. 2 of 3
Shipment 792
Pack. 3 of 3
To: Yishay
From: Vered
To: Yishay
From: Vered
To: Yishay
From: Vered
Handling
To: Boston
From: TLV
Shipment 792
Pack. 3 of 3
Routing
To: Yishay
From: Vered
To:
Boston
To: Boston
From: TLV
Shipment 792
Pack. 3 of 3
To: Yishay
From: Vered
Transport
Haifa
JFK
TLV
BGN

2. JFK N.Y. To: Yishay From: Vered Boston Shipment 792 Shipment 792
Pack. 1 of 3
Shipment 792
Pack. 2 of 3
Shipment 792
Pack. 3 of 3
To: Yishay
From: Vered
To: Yishay
From: Vered
To: Yishay
From: Vered
To: Boston
From: TLV
Shipment 792
Pack. 3 of 3
To: Yishay
From: Vered
To:
Boston
To: Boston
From: TLV
Shipment 792
Pack. 3 of 3
To: Yishay
From: Vered
JFK
N.Y.
Boston

3. Example Send a file over the internet packet link (fixed rate) Modem
card
buffer

4. Delay Models place Computation transmission propagation (Queuing) C B
time

5. Queuing Theory
The theoretical study of waiting lines, expressed in mathematical terms
input
output
server
queue
Delay= queue time +service time

6. Describe the dynamics of the system Evaluate its Performance
The Problem
Given
One or more servers that render the service
A (possibly infinite) pool of customers
Some description of the arrival and service processes.
Describe the dynamics of the system Evaluate its Performance
If there is more than one queue for the server(s), there may also be some policy regarding queue changes for the customers.

7. (all the information necessary to completely describe the system)
Common Assumptions
The queue is FCFS (FIFO).
We look at steady state : after the system has started up and things have settled down.
State=a vector indicating the total # of customers in each queue at a particular time instant
(all the information necessary to completely describe the system)

8. Notation for queuing systems
omitted if infinite
:Where A and B can be
D for Deterministic distribution
M for Markovian (exponential) distribution
G for General (arbitrary) distribution

9. The M/M/1 System
Poisson Process
Exponential server
output
queue

10. Arrivals follow a Poisson process
Readily amenable for analysis
Reasonable for a wide variety of situations
a(t) = # of arrivals in time interval [0,t]
 = mean arrival rate
t = k ; k = 0,1,…. ; 0
Pr(exactly 1 arrival in [t,t+]) = 
Pr(no arrivals in [t,t+]) = 1-
Pr(more than 1 arrival in [t,t+]) = 0
Pr(a(t) = n) = e- t ( t)n/n!

11. Model for Interarrivals and Service times
Customers arrive at times t0 < t1 < Poisson distributed
The differences between consecutive arrivals are the interarrival times : n = tn - t n-1
n in Poisson process with mean arrival rate , are exponentially distributed,
Pr(n  t) = 1 - e- t
Service times are exponentially distributed, with mean service rate :
Pr(Sn  s) = 1 - e-s

12. System Features  This is a Markovian System
Service times are independent
service times are independent of the arrivals
Both inter-arrival and service times are memoryless
Pr(Tn > t0+t | Tn> t0) = Pr(Tn  t)
future events depend only on the present state
 This is a Markovian System

13. Discrete Time Markov Nk =# at time k
Pri,j=Pr(Nk+1= j | Nk= i) = probability to go from i to j events in the time interval 
Pr0,0 1-  
Pri,i+1   
Pri,i-1   
Pri,i  1- -  
Pri,j  0
1-  - 
1-  
1-  - 
 
  1 2
 
 

14. Steady State Solution # transitions kk+1 = # transitions k+1 k
therefore:
Prk ( )= Prk+1 ( )
Prk+1= (/) Prk = Prk  Prk= kPr0
and since:
Prk=(1-) k

15. M/M/1 Performance
Average # in the queue:
Variance # in the queue

16. M/M/1 Performance
is the mean waiting time to get through the queuing system
Little’s Law : the mean # of customers in the system is 

17. A Heuristic Proof for Little’s law
N(t)=A(t)-D(t)

18. Service Times Distributions for single server systems
when service times (X1,X2,….) are i.i.d and independent of the interarrival times,
the average service time:
the expected customer waiting time in queue:
(P-K formula)
is the second moment
for M/M/1 :
for M/D/1 :

19. Service Times Distributions for single server systems (cont.)
the total waiting time (in queue+in service):
The expected # of customers in the system (Little’s law) :
The expected # of customers in the queue

20. Networks of queues Interaction of communication lines
transmission from one channel to another via junction
useful application: estimate delay of packets in a packet switch.
Problem: the interarrival times at the second queue depend on the packet lengths

21. Queuing networks : Model assumptions
M/M/1 servers only ( , )
the output of each server is a Poisson similar to its input (Burke)
the sum of several inputs I is a Poisson
packets length is independent random (Kleinrock) 1
1+ 2+ 3 2 3