Delay Analysis of Three-State Markov Channels

1Jian Zhang, 1Zhiqiang Zhou, 2Tony T. Lee, and 1Tong Ye

1State Key Lab of Advanced Optical Communications and NetworksShanghai Jiao Tong University, Shanghai, China

2Chinese University of Hong Kong (Shenzhen), Shenzhen, China

Aug. 22, 2017 @ QTNA 2017

Outline

n Background and Problemn Hybrid embedded Markov Chainn Mean waiting timen Conclusion

Background

n Wireless channel suffers from slow/fast fading and transmission rate changes during packet service

n Our goal is to figure out what factors affect delay performance

Three-state Markov Channel Model

n Rate fluctuation of wireless channel can be capture by a three-stage Markov Chainn 𝜇": service rate in state j (𝜇# = 0, 𝜇' < 𝜇))n 𝜋": Steady state probability that the channel is in state 𝑗

𝑸 =−𝑓# 𝑓# 0𝑓',# −𝑓' 𝑓',)0 𝑓) −𝑓)

𝜋# =𝑓',#𝑓)

𝑓#𝑓) + 𝑓)𝑓',# + 𝑓#𝑓',)𝜋' =

𝑓#𝑓)𝑓#𝑓) + 𝑓)𝑓',# + 𝑓#𝑓',)

𝜋) =𝑓#𝑓)

𝑓#𝑓) + 𝑓)𝑓',# + 𝑓#𝑓',)

𝝅𝑸 = 𝟎⟹

3𝜋44

= 10

crash1bad

f0

f1,0

2good

f1,2

f2

𝑫 =𝜇# 0 00 𝜇' 00 0 𝜇)

M/MMSP/1 queuing model

n If traffic input is a Poisson traffic with rate 𝜆, wireless communication system can be by an M/MMSP/1 modeln 𝑋 𝑡 , 𝑌 𝑡 : (number of packets, channel state) at time 𝑡

...0,1 1,1 n-1,1 n,1

...0,2 1,2 n-1,2 n,2

...0,0 1,0 n-1,0 n,0

f0

f2 f1,2

f1,0f0

f2 f1,2

f1,0 f0

f2 f1,2

f1,0f0

f2 f1,2

f1,0

λ λ λ λ

λ λ λ λ

µ1 µ1 µ1 µ1

λ λ λ λ

µ2 µ2 µ2µ2

...

...

...

Steady state probability𝑝<," = lim

@→B𝑃{𝑋 𝑡 = 𝑛, 𝑌 𝑡 = 𝑗}⟹

⟹ Mean Delay: An numerical Solution

Outline

n Background and Problemn Hybrid embedded Markov Chainn Mean waiting timen Conclusion

Feature of M/MMSP/1 Queue

n Service time of packets are not independently and identically distributed. It depends on the state in which the packet start being served [1].n Start service in larger service rate state will lead to a smaller service

time.n The start service state of one packet is dependent on the start service

state of last packet.

n We use hybrid embedded Markov chain to describe the channel state transition during the service of packets.

[1] Mahabhashyam, S R., Natarajan G.: On queues with Markov modulated service rates. Queueing Systems 51(1), 89-113(2005).

Hybrid Embedded Points

n 𝛷": Epoch when state transits, after which channel state is 𝑗n 𝑆": Epoch when service starts, after which channel state is 𝑗

n If current epoch is an embedded point with state 𝑗n Probability that next embedded point is 𝛷4(𝑖 ≠ 𝑗): 𝑓",4/(𝜇" + 𝑓");n Probability that next embedded point is 𝑆": 𝜇"/(𝜇" + 𝑓");n Holding time, time from current embedded point to next embedded point,

follows an exponential distribution with parameter𝜇" + 𝑓" .

Current Epocht

T2 (µj)

T1 (fj)

Φi Sj

Probability at Embedded Points

n 𝜋O<," 𝑚 = Pr{start−service state of 𝑚@S packet is 𝑗|an arrival sees 𝑛 packets in the system}

n 𝜑O<," 𝑚 = Pr{during service of 𝑚@S packet, channel transits into state 𝑗|an arrival sees 𝑛 packets in the system}

n-1 ...n m m-1 01... 2t

in servicearrival

Start Service Probability

n State equations of conditional start service probabilityn 𝜋O<,"(𝑚) =

UVUVWXV

𝜋O<," 𝑚 − 1 + 𝜑O<,"(𝑚 − 1)

n 𝜑O<,"(𝑚) = ∑ XZ,VUZWXZ

𝜋O<,4 𝑚 + 𝜑O<,4(𝑚)4["

n Matrix form of conditional start service probability𝜋O< 𝑚 = 𝐷 −𝑄 ^'𝐷

_𝜋O< 𝑚 − 1

where 𝜋O< 𝑚 = 𝜋O<,# 𝑚 , 𝜋O<,' 𝑚 , 𝜋O<,) 𝑚_

n Start service probabilityn 𝜋O" = ∑ 𝑝<B

<`# 𝜋a<," 𝑛

Outline

n Background and Problemn Hybrid embedded Markov Chainn Mean waiting timen Conclusion

Residual waiting time at embedded points

n A new arrival packet sees 𝑛 packets in the buffer:n 𝑊<," 𝑘 : The residual waiting time from the epoch when this new arrival

becomes the𝑘@Spacket in the buffer while the channel is in state𝑗to theepoch when it becomes an HOL packet.

n 𝑉<," 𝑘 : The residual waiting time from the epoch when the channeltransits to state𝑗while this new packet is the𝑘@Spacket in the buffer tothe epoch when it becomes an HOL packet.

n-1 ...n 013 2t

in servicearrival

t

... 013 2

start service

kt

Mean Waiting Time

n State equations of conditional waiting time

n 𝑊<," 𝑘 + 1 = UVUVWXV

'UVWXV

+𝑊<," 𝑘 + ∑ XV,ZUVWXV

'UVWXV

+ 𝑉<,4 𝑘 + 14["

n 𝑉<," 𝑘 + 1 = UVUVWXV

'UVWXV

+𝑊<," 𝑘 + ∑ XV,ZUVWXV4["

'UVWXV

+ 𝑉<,4 𝑘 + 1

n Matrix form of conditional waiting time𝑾< 𝑘 + 1 = 𝑫 −𝑸 ^'𝑫𝑾< 𝑘 + 𝑫 − 𝑸 ^'𝟏

where𝑾< 𝑘 = 𝑊<,# 𝑘 ,𝑊<,' 𝑘 ,𝑊<,) 𝑘

_

Mean Waiting Time

n Mean waiting time n 𝑊 = ∑ ∑ 𝑊<," 𝑛 𝑝<,"B

<`')"`#

=ghaij _ W g

gkl∑ j _V mV^maV ^

lgkl

gno

mo^maopVqo

'^rha

n 𝛽 = 𝜇'𝜇)/(𝜇'𝜇) + 𝜇'𝑓) + 𝜇)𝑓',)) is one of the 3 eigenvalues of 𝑫 −𝑸 ^'𝑫.n 𝛽 → 0: state transition rate is much larger than service rate.n 𝛽 → 1: state transition rate is much smaller than service rate.

Simulation result

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9100

101

102

103

M

ean

Wai

ting

Tim

e

Arrival Rate λ

f=0.001 f=0.01 f=0.1 f=1 M/M/1

µ0=0, µ1=0.5, µ2=1, f0=4f, f10=f, f12=7.5f, f2=2f

Conclusions

n Many problems in communication and computer networks can be modeled as M/MMSP/1 queueing model with several states, of which the numerical solution provides little physical insight

n With the help of hybrid embedded Markov chain, we obtain a structural solution and find that delay is influenced by state transition rate significantly

n Our approach can be easily extended to other finite-state M/MMSP/1 queueing model

Thank You!