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The M/M/1 Queueing System

- The M/M/1 Queueing System consisits of a single queueing station with a single server. The name M/M/1 reflects standard queueing theory nomenclature whereby:
- The first letter indicates the nature of the arrival process.
- e.g., M stands for memoryless, which here means a Poisson process, G stands for a general distribution of interarrival time, D stands for deterministic interarrival times.
- The second letter indicates the nature of the probability distribution of the service times.
- The last number indicates the number of servers.

The M/M/1 Queueing System

- We have already established, via Little’s Theorem, the relations

Between the basic quantities,

N = Average number of customers in the system

T = Average customer time in the system

NQ = Average number of customers waiting in queue

W = Average customer waiting time in queue

- Given these statistics, we will be able to derive the steady-state probabilities
- pn= Probability of n customers in the system, n = 0,1, …
- From these probabilities, we can get

and using Little’s Theorem,

Arrival statistics—the Poisson process

- A stochastic process taking nonnegative integer values is said to be a Poisson process with rate λif
- A(t) is a counting process that represents the total number of arrivals that have occurred from 0 to t, and for s < t, A(t)-A(s) equals the numbers of arrivals in the interval (s, t].
- The numbers of arrivals that occur in disjoint time intervals are independent.
- The number of arrivals in any interval of length τ is Poisson distributed with parameter . That is, for all t, τ> 0,

Arrival statistics—the Poisson process (Cont)

- We list some of the properties of the Poisson process that will be of interest:
- Interarrival times are independent and exponentially distributed with parameter λ; that is, if tndenotes the time of the ntharrival, the intervals have the probability distribution

and are mutually independent.

- For every and ,

where we generically denote by o(δ) a function of δsuch that

Arrival statistics—the Poisson process (Cont)

- We list some of the properties of the Poisson process that will be of interest:
- If two or more independent Poisson processes are merged into a single process , the latter process is Poisson with a rate equal to the sum of the rates of its components.
- If a Poisson process is split into two other processes by independently assigning each arrival to the first (second) of these processes with probability p ( 1-p, respectively), the two arrival processes thus obtained are Poisson.

Service Statistics

- Our assumption regarding the service process is that the customer service times have an exponential distribution with parameter μ, that is, if sn is the service time of the nth customer,
- An important fact regarding the exponential distribution is its memoryless character, which can be expressed as

for the interarrival and service times and ,respectively

- Verification of the memoryless property follows the calculation

Markov Chain Formulation

- Let us focus attention at the times

where δis a small positive number. We denote

Nk= Number of customers in the system at time

- Let denote the corresponding transition probabilities
- We can show that

Derivation of the Stationary Distribution

- Consider now the steady-state probabilities
- The probability that the system is in state n and makes a transition to n+1 in the next transition is the same as the probability that the system is in state n+1 and makes a transition to n, that is,

Derivation of the Stationary Distribution (Cont.)

- By taking the limit in the equation aswe obtain

These equations can also be written as

where

- It follows that
- If , the probabilities pn are all positive and add up to unity, so
- Combing the last two equations, we finally obtain

Derivation of the Stationary Distribution (Cont.)

- We can now calculate the average number of customers in the system in steady-state:

and finally, using , we have

- The average delay per customer is given by Little’s Theorem,

Using , this becomes

Derivation of the Stationary Distribution (Cont.)

Average Number in the system N

- The average waiting time in queue, W, is the average delay T less the average service time 1/μ, so
- By Little’s Theorem, the average number of customers in queue is

Utilization Factor

Occupancy Distribution upon Arrival

- The steady-state occupancy probabilities upon arrival,

need not be equal to the corresponding unconditional steady-state probabilities,

- It turn out, however, that for the M/M/1 system, we have

Occupancy Distribution upon Arrival (Cont.)

- A formal proof under the preceding assumption:
- Let
- We have, using Bayes’ rule,
- By assumption, the event A(t, t+δ) is independent of the number in the system at time t, therefore,

and we obtain

Occupancy Distribution upon Departure

- Let us consider the distribution of the number of customers in the system just after a departure has occurred, that is, the probabilities
- The corresponding steady-state values are denoted
- It turns out that

The M/G/1 System

- Let
- The Pollaczek-Khinchin(P-K) formula:

where W is the expected customer waiting time in queue and

- The total waiting time, in queue and in service, is

M/G/1: System (Cont.)

- Appling Little’s formula to W and T, we get the expected number of customers in the queue NQ and the expected number in the system N:
- Under exponential service time, i.e., ,
- When service time is deterministic, i.e.,

M/G/1: System (Cont.)

- Denote
- = Waiting time in queue of the ith customer
- = Residual service time seen by the ith customer. By this

we mean that the customer j is already being serve

when i arrives, is the remaining time until customer

j’s service time is complete. If no customer is in

service(i.e., the system is empty when i arrives), then

is zero

- = Service time of the ith customer
- = Number of customers found waiting in queue by the ith

customer upon arrival

- We have

M/G/1: System (Cont.)

- By taking expectations and using the independence of the random variableand, we have
- Taking the limit as , we obtain

where

R = Mean residual time, define as

M/G/1: System (Cont.)

Residual Service time γ（τ）

- By Little’s Theorem, we have

and by substitution in the waiting time formula, we obtain

where is the utilization factor; so finally,

M/G/1: System (Cont.)

- The time average of in the interval [0,t] is

where M(t) is the number of service completions within[0,t], and Xiis the service time of the ith customer.

- We can also write this equation as

and assuming the limits below exist, we obtain

- Assuming that time averages can be replaced by ensemble averages, we obtain
- The P-K formula,

Crossbar Switches

- Crossbar switches are an important general architecture for fast switches.
- 2 x 2 Crossbar Switches
- A general N x N crossbar switch

Input Queueing -- "If we come in together then we wait together"

Output Queueing -- "We wait at the destination (output) together"

Input Queueing versus Output QueueingThe queueing will be at the input or at the output?

- The switch fabric speed is equal to the input line speed
- To avoid collision on the single speed switch fabric, only one input line can can place a packet on the switch fabric at a time. This requires the other inputs to stop the packet from entering the switch fabric. This is implemented using an queue at the input.
- The switch fabric speed is N times faster than the input line speed
- The internal switch has slot times which are N times as fast as those of the input lines. The packets enter the crossbar switch together and are shifted to the outputs together. This requires queueing at the outputs to avoid collisions.

General Assumptions for Analysis

- In any given time slot, the probability that a packet will arrive on a particular input is p. Thus p represents the average utilization of each input.
- Each packet has equal probability 1/N of being addressed to any given output, and successive packets are independent.

1

1

1

1

- Switch with Speedup factor of N.
- Arriving packets reach the targeted output ”immediately”.
- = # arriving packets at the tagged queue during a given time slot m

Analysis of Output Queueing

as N

Poisson Distribution.

Analysis of the Output Queue Size

: the number of packets in the tagged queue at the end of the time slot m

Using a standard approach in queueing analysis

The mean queue size for an M/D/1 queue

The mean stead-state queue size

As

Analysis of the Packet Waiting Time

The time slots that packet must wait while packets that arrived in earlier time slots are transmitted

The time slots that packet must wait additionally until it is randomly selected out of the packet arrivals in the time slot m

Analysis of the Packet Waiting Time

- b: the size of the batch the packet arrives in

Analysis of the Packet Waiting Time

- the mean steady-state waiting time

The mean waiting time for an M/D/1 queue

Analysis of the Packet Waiting Time

- The steady-state waiting time probabilities:

Analysis of Input Queueing

- More complex system to analyze than output queueing case.
- In order to analyze it, we make a simplifying assumption of "heavy load", i.e. all queues are always full.
- This is a worst-case assumption.

head-of-line (HOL) blocking

Winning packet

Input Queues

Outputs

Losing packet

3

1

1

2

1

Internally Nonblocking Switch

2

cannot access output 2 because it is blocked by the first packet

4

3

3

4

Internally Nonblocking Switch:Dropping packets

0= Pr[ carry a packet ]

1

2

3

2

Pr[ carry a packet ] =p

for large N

For p=1, 0= 0.632

the fictitious output queues used for analysis

Output 1

Fictitious Output Queues formed by HOL packets

(3,2)

(1,2)

Output 2

(2,3)

Output 3

(4,4)

Output 4

(input, output)

Outputs

(1,1)

(1,2)

1

(2,1)

(2,3)

Internally Nonblocking Switch

2

(3,2)

(3,2)

3

(4,1)

(4,4)

4

2

3

4

5

*

0.75

0.68

0.66

0.64

- How about small N?

* : the maximum throughput with input queueing

- Simulation Results with

Large N

Throughout of Input-Buffered Switch

- Consider a fictitious queue i

= # packets at start of time slot m.

= # packets arriving at start of time slot m.

- is Poisson and independent of

as N

For finite buffer size, if p0 > p* = 0.586 at least (p0 - p*)/ p0 fraction of packets are dropped.

Must keep p0 < p*

Meaning of Saturation Throughput

p0 = p = throughput

Queuing scenario for the delay analysis of the input-buffered switch

Fictitious Queues

Output 1

Input Queue

1/N

1/N

N

2

Output 2

HOL

1/N

Time spent in HOL are independent for successive packets when N is large

Output N

Service times at different fictitious queues are independent

The busy periods and interpretations for delay analysis of an input queue

U(t)

X3

X0

X1

X2

X0

t

Idle period

Y

Busy period

Busy period

Arrivals here are considered as arrivals in intervals i-2

Arrivals here are considered as arrivals in intervals i-1

Xi-1

Xi

Illustration of the meanings of random variables used in the delay analysis of an input queue

mi =2 prior arrivals

Arrival of the packet of focus. One simultaneous arrival to be served before the packet; L=1.

Departure of packet of focus.

(1)

(1)

(2)

Xi

Xi+1

Ri

W

-- Packet arrival in interval i.

-- packet departure in interval i+1.

(n)

-- number of arrivals

Three Selection Policies

- Random Selection Policy
- If k packets are addressed to a particular output, one of the k packets is chosen at random, each selected with equal probability 1/k.
- Longest Queue Selection Policy
- The controller sends the packet from the longest queue
- Fixed Priority Selection Policy
- The N inputs have fixed priority levels and of the k packets, the controller send the one with highest priority

Different contention-resolution policies have different waiting time versus load relationships, but a common maximum load at which waiting time goes to infinity.

_

W

p0

Conclusion

- Mean queue length are always greater for queueing on inputs than on outputs
- Output queues saturate only as the utilization approaches unity
- Input queues saturate at a utilization that depends on N, but is approximately 0.586 when N is large

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