Download Capacity Assignment in Centralized Networks: 2) Definition of

Capacity Assignment in Centralized Networks: 2) Definition of Network Problem: Given network data:*The network
topology is given: the locations of the terminals, links and central processor are given.*In the simplest case, we assume that
link cost is proportional to capacity. *A budget to acquire link capacities. This implies that total link capacity is also given.
Why? *3)Given Network Data Cont *The network traffic statistics (requirements) are given; i.e., we know: **The
average message lengths **Average arrival rates of messages: number of messages flowing between any two points in the
network. *4) Task: Assign a capacity (in bps) to each link in the network to provide a specified level of service to users (or
any other network performance criteria). 5) Network performance criteria *There are two main criteria to evaluate the
quality of service to users: *Minimize the average message time delay in the network. *Minimize the largest time delay
expected in any link in the network. 6)Notes: *Most of the discussion here is based on the pioneering work of L.
Kleinrock. *The examples in this lecture are taken from Schwartz. 7) Example: Suppose that there are seven cities, each
with a specified number of terminals (listed below) to be connected to a central computer in Washington, D.C.: *Chicago
10 terminals *Detroit 9 *Charlotte, NC 4 *Miami, Fla 6 *New Orleans 6 *N.Y. 12 *Columbus 4 8) Star Topology
Network: *Let’s assume that network has a star topology as shown below. Assume that each terminal produces a
message, on average, once every 30 seconds and that the average message length is 120 bits. *At each city node there is a
concentrator that is used to combine incoming messages from terminals and route them over the appropriate outgoing link
after some necessary processing and buffering. 10) Let *i = average message rate (messages/sec) for link i. *Ci = capacity
(in bps) for link i. *The job is to determine the capacity (in bps) to allocate to each of the seven links in the network. 11)
We assume that nodal processing delay is negligible compared to the queuing delay incurred by messages waiting for the
outgoing link to be made available. 12) *The average arrival message rate for link i (that is the average number of
messages per second arriving at link i ) is the sum of all incoming messages routed over that link. *In the example, all the
terminals have the same message rate (1/30 messages/sec). Therefore, 1 = 10* 1/30 =0.33 messages/sec, *2 = 9* 1/30
=0.3 messages/sec, etc. 13) Message arrival rates for the example PICTURE 14) We make the following additional
assumptions necessary to model the telecommunication network as a network of independent M/M/1 queues: *A Poisson
message arrival rate with i messages/sec arriving on average at link i. *An exponential distribution of message lengths
with 1/i bits/message on average for link i. *Infinitely large buffer capacity. Messages arrive independently. 15) *Under
these assumptions and by applying queuing theory, the average time queuing delay in seconds incurred by messages at link
i is given by: *Ti= 1/(iCi -i), where Ci is the capacity of link i that needs to determined. *This delay includes the average
time taken to transmit a message (1/(iCi) sec) plus the message buffering delay. *i = i/(iCi) is called the traffic intensity
(or utilization factor) for link i. It is value is always less than 1. 16) Ti can be defined in terms of i, Ti = 1/iCi(1-i) So, Ti
= 1/iCi * 1/(1-i) ** Therefore, Ti increases with: **Traffic intensity (i). **The increase in the average time required to
transmit a message 1/(iCi ). *In our example, the average message length is 1/i = 120 bits/message for each link. 17)
Network Design Objective: *The design objective is to minimize the time queuing delay averaged over the entire network
with total capacity of the links assumed fixed. (This also means that the budget for acquiring capacity is fixed because we
assume that cost is linearly proportional to capacity). 18) *Let  be the total incoming message rate for the network. *In
our example, what is the value  of ? *  = 0.333+0.3+0.4+0.2+0.133+0.133+0.2=1.7 *Using queuing theory, the average
message delay in the network is defined as T=1/ (i=1 to n) i * Ti. 19) Assuming that the total capacity is held fixed,
C=(i=1 to n) Ci, we want to determine the capacity of each link Ci in order to minimize T. Min T subject to C=(i=1 to n) Ci.
20)If we solve this minimization problem, we obtain the optimal capacity of link i: NOTE  is a square root. CI
=(i/i)+[C(1-)(i/i)]/[ (i=1 to n) (i/i)], where =(i/i)/C plays the role of traffic intensity for the entire network.
The value of  can be easily computed because C is assumed to be known. 21) *This formulae is called the square root
assignment rule. *The square root assignment rule first assigns the absolute minimum capacity to each link i (that is, i/i)
and then allocates the remaining capacity to each link following a square root assignment strategy. *The minimum average
message delay for the entire network is given by: T*=[(i=1 to n) (i/i)]2/C(1-) NOTE: Add up all square roots, then
square the sum. 22) *Let's apply this rule to the example. *Here i =  = 1/120 for all links because *1/i = 120
bits/message. C=(i=1 to n) (i/i)=238.8bps. For our example, let's say the budget allows to acquire a total capacity of 4500
bps for all links in the network, that is C=4500 bps.