Introduction: Two motivating examples for the analytical approach

Transcription

1 Introduction: Two motivating examples for the analytical approach Hongwei Zhang Acknowledgement: this lecture is partially based on the slides of Dr. D. Manjunath

2 Outline Two example systems Efficient transport of packet voice calls Achievable throughput in an input-queued packet switch Importance of quantitative modeling Summary

3 Outline Two example systems Efficient transport of packet voice calls Achievable throughput in an input-queued packet switch Importance of quantitative modeling Summary

4 Quantitative Models in Networking Two examples to illustrate insights from quantitative models and their analysis. Multiplexing packet voice on a high speed communication channel. Asymptotic analysis of the saturation throughput of an input queued cell switch

5 Outline Two example systems Efficient transport of packet voice calls Achievable throughput in an input-queued packet switch Importance of quantitative modeling Summary

6 Digital voice Analog voice is sampled at 8K samples/sec; At 8bits/sample the bit rate is 64Kbps Sophisticated coding can lower bit rates; 32 Kbps, 16 Kbps, 8 Kbps, 6.4 Kbps and even lower values is possible and is routinely used Samples are obtained at regular intervals Layered or embedded coding allows bit rate to be lowered to reduce traffic load In adaptation to network properties (e.g., packet loss in wireless networks)

7 Digital voice: quality metrics Distortion: Faithfulness in the reproduction of the source waveform at the destination. Delay: Excessive delay between generating and playout is not acceptable for interactive conversation.

8 Packet voice Make packet from samples of fixed intervals Typical packetisation intervals are 10/20 ms Form a packet by adding header and trailer and send it over network Can suppress (i.e. not generate or send) packets during silence periods Voice activity at a source is a random phenomena; Sample, hence packet, generation instants at source will be a random process.

9 Packet voice (contd.) Many such random processes, corresponding to the number of sources sharing the link at a given time, are superposed at the multiplexer Packets can experience random loss (causing distortion) and/or random delay Measures of packet voice performance: Packet Delay, Packet Loss Rate

10 Packet Voice Multiplexing System A LAN-WAN packet router multiplexes several digitised, coded and packetised voice sources into an interlocation link

11 Abstraction: queuing model for studying the performance of packet voice multiplexing in the system Channel capacity B C Buffer threshold that should be exceeded rarely to avoid long delay

12 Abstraction Summary One link of capacity C bits/sec multiplexing many calls between two destinations Input to the multiplexer is a random stream of voice packets from each source

13 Quality Constraints Distortion constraint: Deliver at least a fraction α of the bits from a source α = 0.95 implies that for every sample of, say, 4 bits, on an average 3.98 bits should be delivered to destination. Delay constraint: Do not delay a packet by more than δ at the multiplexer. δ = B/C. Hence, δ can be specified as the maximum occupancy of the buffer

14 Design Problem Evaluate design choices by determining the number of calls that can be simultaneously carried subject to distortion and delay constraints being satisfied

15 Design #1 : Bit dropping at multiplexer Let B = δ * C. Not allowing buffer occupancy to ever exceed B will satisfy delay constraint. Drop bits that are in excess of B Assume very intelligent dropping at multiplexer to ensure quality degradation is evenly spread among sources use layered coding Design question reduces to: Find number of sources (N 1 ) such that P{a bit is dropped} < 1-α Closed-loop control

16 Bit Dropping at the Multiplexer bits dropped B 0 time Evolution of the multiplexer buffer level with bit dropping control

17 Design #2: lower coding rate at sources Ask sources to code at α*(full rate); Ensure no bits are dropped by multiplexer Delay constraint is achieved by choosing the number of sources (N 2 ) such that buffer occupancy never exceeds B Due to randomness in arrival processes, it is impossible to have absolutely no bits dropped at the multiplexer; Thus translate last condition to requiring P{an arriving bit sees an occupancy>b} < ε with ε = Open-loop control

18 Analysis

19 Analysis (contd.)

20 Bit-dropping analysis Stationary distribution of buffer level with bit dropping is supported on interval [0, B], with a point mass z at 0 That is, for 0 < x < B (note strict inequalities) Derivation in Appendix D.1.9, page 841 or R0

21 Bit-dropping analysis (contd.)

22 The bit dropping design problem }

23 Low rate coding analysis Cannot drop any bits in multiplexer to satisfy delay constraint Because sources are already coding at lowest rate every bit should be carried to satisfy distortion constraint We will have to have infinite buffers, i.e., B =

24 Distribution function can be obtained from the one for bitdropping by letting mean packet length be α/µ and letting B Low rate coding analysis (contd.)

25 Low rate coding analysis (contd.) }

26 Results bit-dropping low-bit-rate coding delay bound (in packet transmission times) Maximum offered load that meets performance objectives for bit dropping and low bit rate coding, plotted vs. the delay objective expressed as µb

27 Discussion Bit dropping at multiplexer accommodates up to twice the number of calls that lower source coding can accommodate Reasoning: Bit dropping drops only when necessary while Source Coding is very conservative Caveat: Model is too simplistic; many practical details are glossed over; But intuition is gained! This analysis helped in design of Integrated Access Terminal for Wideband Packet Networking by AT&T in 1990

28 Outline Two example systems Efficient transport of packet voice calls Achievable throughput in an input-queued packet switch Importance of quantitative modeling Summary

29 Achievable throughput in an input-queuing packet switch

30 Packet switches Packet switch moves packets (a group of bits) from an input on which it arrives at a switch, to an output specified by packet header Example packet switches: Ethernet switches and routers Consider packet switches operating on fixed size packets called cells Many new high speed routers use such switches internally Time slotted operation of the switch Slot length=cell transmission time

31 Queueing in cell switches Non Blocking Switch: If set of destinations of cells at input is a permutation (destinations are unique), all cells can be transferred to respective outputs simultaneously. In each slot at most one cell arrives to input i and has a random destination d i Destination conflict: In a slot, more than one cell wants to go to same output Cells need to be queued at either input (Input Queueing) or output (Output Queueing)

32 Output queued switches All cells for a destination are sent to the output One cell is transmitted on the output line, and others are queued Potentially good performance output is never idle if there is a cell wanting to leave on that port Scalability problem: Switch should be capable of moving up to N cells to an output in a slot N: # of input ports More importantly, memory should be capable of supporting such read-write speed Improvements in memory access speed have not kept pace with that of processor technologies or even with the size of memory

33 Input queued switches Arriving cells are placed in input FIFO queue Cell at HOL (for Head of Line) is sent to output if it is free If more than one HOL cell wants to go to same output only one is sent HOL blocking: cells behind HOL cell can be blocked if destination of HOL cell is not free Switch should be able to move exactly one cell from each input (to each output) in a slot No scalability problem but a possible performance problem due to HOL blocking

34 HOL blocking in IQ switches

35 Saturation throughput analysis Consider an N*N switch Assumptions: The inputs are always saturated: departing cell is immediately replaced by a fresh cell Any of the N output ports is equally likely to be the destination of this cell and is independent of everything else (e.g., destination of other cells)

36 Saturation throughput analysis (contd.)

37 Markov Chain for the 2*2 switch

38 Results

39 Results (contd.)

40 Discussion

41 Discussion: VOQ 1996: virtual output queueing (VOQ) to address the HOL blocking problem, achieving a throughput of 1.0 per port Each input i maintains a separate queue of cells for each output j: VOQ ij Scheduling algorithm Creating a bipartite graph with edges between input and output, each edge (i, j) is assigned a weight equal to the number of cells in VOQ ij Maximum-weight matching algorithm is executed at the beginning of each slot to find which HOL cells will be transmitted

42 VOQ (contd.) VOQ 12 VOQ 21 VOQ 22 VOQ 11 Q Q 12 2*2 switch Q Q 22

43 Discussion: PRN Designing an IQ switch with capacity 1.0 is identical to a problem in packet radio network (PRN) scheduling problem: Which nodes should transmit simultaneously to achieve maximum throughput in the network 2 Q 32 3 Q 43 1 Q 12 4 Q 54 Q 81 Q 27 Q 37 8 Q Q 76 Q 65 6

44 Outline Two example systems Efficient transport of packet voice calls Achievable throughput in an input-queued packet switch Importance of quantitative modeling Summary

45 Importance of quantitative modeling Two very different kinds of questions for two very different networking subsystems In practice, Toy models for insight Abstract many details, amenable to mathematical analysis, provable results and possibly closed form expressions These models can be used to drive control algorithms and scheduling policies and to motivate heuristics More detailed models for simulation and/or numerical analysis simple, toy models can validate computer simulations, and algorithms and heuristics derived from simple models can be experimented with more detailed (simulation/numerical) models

46 Importance of quantitative modeling (contd.) Some success stories from before Teletraffic Engineering of the telephone network The Erlang-B formula for call blocking probability in a circuit switched network Dynamic Non Hierarchical Routing where excess capacity available was utilized in these networks to significantly improve the traffic carrying capability

47 Importance of quantitative modeling (contd.) More recent successes Network calculus, and its use in end-to-end deterministic analysis of the performance of network flows and in design with worst-case performance guarantees Effective bandwidths approach for link sizing with stochastic performance guarantees Distributed asynchronous algorithms for rate allocation in achieving global fairness for elastic flows

48 Importance of quantitative modeling (contd.) In networking, unfortunately, designs are not derived from models Simple (not naive!) technologies are first developed and quickly deployed to provide simple services To provide sophisticated services these simple designs need to be understood and hence analyzed, especially the interactions Examples: Aloha, Ethernet, TCP and their analysis has led to new algorithms for use with new technologies

49 Outline Two example systems Efficient transport of packet voice calls Achievable throughput in an input-queued packet switch Importance of quantitative modeling Summary

50 Summary Two example systems Efficient transport of packet voice calls Achievable throughput in an input-queued packet switch Importance of quantitative modeling

51 Homework #1 Chapter 1 of R0 Problem 1.3 Total points: 100