An unreliable vacation queueing model and its application ... · PDF fileIntroduction Model Mathematical analysis Performance metrics Numerical results References An unreliable vacation

Introduction Model Mathematical analysis Performance metrics Numerical results References

An unreliable vacation queueing model andits application on the DRX mechanism for

power saving in 3GPP LTE

Ioannis Dimitriou

Dept. of Electrical and Electronic EngineeringIntelligent Systems and Networks GroupImperial College, London SW7 2BT,UK

[email protected]

ICEP 2012


Outline

1 IntroductionThe simplified LTEDRX in 3GPP LTE

2 Model3 Mathematical analysis

General resultsThe imbedded Markov chainWaiting timeDecomposition result

4 Performance metricsPower Saving Factor and Average Power ConsumptionEnergy efficiency and cost function

5 Numerical results6 References


The simplified LTE

The simplified LTE Network Infrastructure

The recent increase of mobile data usage and emergence ofnew applications motivated the 3rd Generation PartnershipProject (3GPP) to work on the Long-Term Evolution (LTE).

LTE is the latest standard in the mobile network technology(succesor of GSM/EDGE and UMTS/HSxPA)

Reduction in the number of network elements, simplerfunctionality.

Peak data rate: UP: 50Mbps, DL: 100 Mbps

OFDMA for DL traffic

SC-FDMA for UL traffic

Advanced antenna techniques (MIMO, Smart antennas)


The simplified LTE

Basic components

IP

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴✤

✤

✤

✤

✤

✤

✤

✤

✤

✤

✤

✤

✤

✤

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

ePC–PDNGW– SGW–

|MME

e-UTRAN(eNB)

❴❴❴

��

�� UE

evolved Packet Core (ePC) network.Packet Data Network Gateway (PDNGW)Service Gateway (SGW)Mobility Management Entity (MME)evolved Universal Terrestrian Radio Access Network(e-UTRAN)evolved NodeB (eNB)User Equipment (UE)


DRX in 3GPP LTE

LTE system will increase power consumption significantly.

Power saving mechanisms are becoming increasinglyimportant for next generation wireless networks.

Idle listening is one of the major source of powerconsumption.

3GPP LTE defines the Discontinuous Reception (DRX)mode.

DRX allows an idle UE to turn off the radio receiver for apredefined period (DRX cycle) instead of continuouslylistening to the radio channel.


DRX in 3GPP LTE

1 Inactivity period: Is activated when eNB buffer becomesempty.

If a packet arrive before inactivity timer expires is forwardedimmediatelly to UE.

2 Short DRX period: Consists of at most N consecutiveshort DRX cycles.

Consists of sleep and on-duration.During sleep-duration, UE turns off its components to savepower.During on-duration, the PDCCH informs UE about buffereddownlink packets destined to UE.

3 Long DRX cycle: If N consecutive short DRX cyclesexpire without arrivals, the UE enters long DRX cycle.

Same operation as short DRX cycle.Long DRX cycle is larger than the short one.The on-period is the same as that in the short one.


Arrivals: Poisson λ(>0).

Service time: Arbitrary distributed, df B(x), pdf b(x), LST

β∗(s), moments b, b(2)

.

Failures: exp(θ).

Recovery: Arbitrary distributed, (R(x), r(x), r∗(s), r , r (2)).With probability p the packet under transmission isdiscarded. With 1 − p, the packet transmission will beinitiated from scratch upon repair completion.

Inactivity timer: Arbitrary distributed, (I(x), i(x), i∗(s), i ,

i(2)

).


Close-down: Arbitrary distributed, (C(x), c(x), c∗(s), c,c(2)).

Start-up: Arbitrary distributed, (S(x), s(x), s∗(s), s, s(2)).

Short DRX period: Consists of at most N consecutiveshort DRX cycles. Each cycle is arbitrary distributed,(V0(x), v0(x), v∗

0 (s), v0, v (2)0 ). The On-period is fixed and

equals τ .

Long DRX period: Each long DRX cycle is arbitrarydistributed, (V1(x), v1(x), v∗

1 (s), v1, v (2)1 ). The On-period is

fixed and equals τ .






General results

Generalized service time

1 A =The elapsed time from the epoch eNB starts thedelivery of a packet until the epoch, it is ready for thedelivery of a "new packet".

2 N(A) = The number of packets that arrive during A.

If aj(t)dt = P(t < A ≤ t + dt ,N(A) = j),a∗(z, s) =

∫∞0 e−st ∑∞

j=0 aj(t)z j ,

a∗(z, s) =β∗(s + θ + λ− λz) + θp 1−β∗(s+θ+λ−λz)

s+θ+λ−λz r∗(s + λ− λz)

1 − θ(1 − p)1−β∗(s+θ+λ−λz)s+θ+λ−λz r∗(s + λ− λz)

.

Let

ρ =∂a∗(z, 0)

∂z|z=1 =

λ(1 − β∗(θ))(1 + θr)θ(β∗(θ) + p(1 − β∗(θ)))


General results

With the above result we can prove the following theorem(simple generalization of Takacs theorem [6]).

TheoremFor (i) ℜ(s) > 0, (ii) ℜ(s) ≥ 0, and ρ > 1 the equation

z − a∗(z, s) = 0, (1)

has one and only one root, z = x(s) say, inside the region|z| < 1. Specifically for s = 0, x(0) is the smallest positive realroot of (1) with x(0) < 1 if ρ > 1 and x(0) = 1 for ρ ≤ 1.


General results

Regeneration cycle

W =the time elapsed from the epoch the inactivity period isenabled until the epoch the next inactivity is about to begin.

If w(t)dt = P(t < W ≤ t + dt),

w∗(s) =λ(1 − i∗(s + λ))x(s)

s + λ+ i∗(s + λ)c∗(s), (2)

where

x(s) = LST of the busy period (only root of z − A(z) = 0),

c∗(s) = c∗(s + λ)v∗(s) + λx(s) (1−c∗(s+λ))s+λ

s∗(s + λ− λx(s)),

v∗(s) = s∗(s + λ− λx(s)){

(1−(v∗

0 (s+λ))N)(v∗

0 (s+λ−λx(s))−v∗

0 (s+λ))

1−v∗

0 (s+λ)

+(v∗

0 (s+λ))N(v∗

1 (s+λ−λx(s))−v∗

1 (s+λ))

1−v∗

1 (s+λ)

}.


General results

The mean duration of regeneration cycle is given by

E(W ) = 1−i∗(λ)λ(1−ρ) + i∗(λ)E(C),

where,E(C) = c∗(λ)E(V ) + (1−c∗(λ))(1+λs)

λ(1−ρ) ,

E(V ) =[s+

(1−(v∗0 (λ))N )v01−v∗0 (λ)

+(v∗0 (λ))N v1

1−v∗1 (λ)]

1−ρ.


The imbedded Markov chain

Tn = nth generalized service completion epoch.Xn = Number of packets in the system just after Tn.

Xn+1 =

Xn − 1 + An+1, if Xn > 0

An+1, if Xn = 0, Qn > 0

Rn+1 + Sn+1 + An+1 − 1, if Xn = 0, Qn = 0, Cn = 0

Sn+1 + An+1, if Xn = 0, Qn = 0, Cn > 0,

An, the number of packets that arrive during a generalizedservice time,Sn, the number of packets that arrive during start up for thetransmission,Cn, the number of packets that arrive during close-downperiod,Rn, the number of packets that arrive during the total DRXcycle,Qn the number of packets that arrive during the inactivityperiod.



{Xn, n = 0, 1, ...} constitutes a Markov chain. Let

πk = P(X = k), k = 0, 1, ... ak = P(A = k), k = 0, 1, ...,qk = P(Q = k), k = 0, 1, ..., ck = P(C = k), k = 0, 1, ....sk = P(S = k), k = 0, 1, ..., rk = P(R = k), k = 1, 2, ....

andΠ(z) =

∑∞k=0 πkzk , A(z) =

∑∞k=0 akzk ,

R(z) =∑∞

k=1 rkzk , S(z) =∑∞

k=0 skzk .

Using the balance equations and forming the pgf’s,

(z−A(z))Π(z) = π0A(z)[z(1−q0)+q0S(z)(z(1−c0)+c0R(z))−1].(3)



where

A(z) = a∗(z, 0), S(z) = s∗(λ− λz), q0 = i∗(λ), c0 = c∗(λ),

R(z) =(1−(v∗

0 (λ))N)(v∗

0 (λ−λz)−v∗

0 (λ))

1−v∗

0 (λ)+

(v∗

0 (λ))N(v∗

1 (λ−λz)−v∗

1 (λ))

1−v∗

1 (λ).

Since (Theorem), z − A(z) never vanishes inside the unit disk,

Π(z) = π0A(z)[z(1−i∗(λ))+i∗(λ)S(z)(z(1−c∗(λ))+c∗(λ)R(z))−1]z−A(z) .

π0 = 1−ρ

1+i∗(λ)[λs+c∗(λ)(E(R)−1)].


Waiting time

Let D be the total waiting time of an arbitrary packet in thesystem (queue+service). Let d(t) its pdf and d∗(s) its LST.Using well-known results,

Π(z) = d∗(λ− λz).

Therefore,

d∗(u) = Π(1 − u/λ) ={

u(1−ρ)A(u)u−λ+λA(u)

}λ(1−F (u))

u(1+i∗(λ)[λs+c∗(λ)(E(R)−1)]),

where A(u) = A(1 − uλ) = a∗(1, u) and F (u) = F (1 − u/λ).


Decomposition result

One of the fundamental results on vacation models isstochastic decomposition.

LemmaFor ρ < 1 the queue length X can be decomposed in twoindependent random variables,

X = Xgb + Xc ,

The pgf’s of the above random variables are

Πgb(z) =(1−ρ)(1−z)A(z)

A(z)−z ,

Πc(z) =1−F (z)

(1−z)(1+i∗(λ)[λs+c∗(λ)(E(R)−1)]).

Clearly Π(z) = Πgb(z)Πc(z).


Mean number of packets in the system.

E(X ) = ρ+ ρ(2)

2(1−ρ)

+i∗(λ)[λ2s(2)+2λs(1+c∗(λ)(E(R)−1))+c∗(λ)

∂2R(z)∂z2 |z=1]

2[1+i∗(λ)[λs+c∗(λ)(E(R)−1)]],

Mean waiting time E(D) = E(X)λ

,

In order to calculate the probabilities of UE receiveractivities we are going to use the Regenerative methodfollowing Ross [4].

Evaluate the mean residence times of UE receiver atinactivity, close-down, start-up, the short and long DRX,listening and busy period, during a regeneration cycle (W ).


Mean residence time in inactivity and close down period

Let I, C and their pdf’s i(t), c(t). Then

i∗(s) = λ1−i∗(s+λ)s+λ

+ i∗(s + λ),

c∗(s) = λ1−c∗(s+λ)s+λ

+ c∗(s + λ).

E(inact) = 1−i∗(λ)λ

and E(cl) = i∗(λ)E(C) = i∗(λ)1−c∗(λ)λ

.


Mean residence time in short DRX period (sleep+On-duration)

Let V0 and its pdf v0(t). Then,

v∗0 (s) = (v∗

0 (s + λ))N +(1−(v∗

0 (s+λ))N)(v∗

0 (s)−v∗

0 (s+λ))

1−v∗

0 (s+λ) .

Provided that inactivity and close-down has expired withoutarrivals,

E(short DRX ) = i∗(λ)c∗(λ)E(V0) =i∗(λ)c∗(λ)(1−(v∗

0 (λ))N)v0

1−v∗

0 (λ).

Assume that there are K short DRX cycles (1 ≤ K ≤ N).

Then we can prove that E(K ) =i∗(λ)c∗(λ)(1−(v∗

0 (λ))N)

1−v∗

0 (λ).


Mean residence time in long DRX period (sleep+On-duration)

Let V1 and its pdf v1(t). Then

v∗1 (s) =

v∗1 (s)− v∗

1 (s + λ)

1 − v∗1 (s + λ)

, E(V1) =v1

1 − v∗1 (λ)

.

Provided that short DRX period has expired (as well asinactivity and close-down) without arrivals,

E(long DRX ) = i∗(λ)c∗(λ)(v∗0 (λ))

NE(V1).

If M the number of long DRX cycles, using Wald’s lemma

E(long DRX ) = E(M)v1 ⇒ E(M) =i∗(λ)c∗(λ)(v∗

0 (λ))N

1 − v∗1 (λ)

.


Mean residence time on start-up period

Start-up period can be enabled during a cycle, only ifinactivity timer expired without arrivals.

If a packet arrives during close down the mean residencetime in start-up period is i∗(λ)(1 − c∗(λ))s.

In case which close-down period expires (w.p. i∗(λ)c∗(λ)),UE enters short DRX. A start-up period is then initiated if apacket arrives either during the short DRX period (w.p.1 − (v∗

0 (λ))N ) or during the long DRX cycle (w.p. (v∗

0 (λ))N ).

Then

E(st) = i∗(λ)s {1 − c∗(λ)

+c∗(λ)[1 − (v∗0 (λ))

N + (v∗0 (λ))

N ]}= i∗(λ)s.


Mean residence time on busy period

if a packet arrives during the inactivity timer (w.p. 1 − i∗(λ))then it is initiated a busy period with one customer. In sucha case E(Binact) =

(1−i∗(λ))ρλ(1−ρ) .

If a packet arrives during close-down period (w.p.i∗(λ)(1 − c∗(λ))), then a start-up period is initiated. ThenE(Bcl) =

i∗(λ)(1−c∗(λ))(1+λs)ρλ(1−ρ) .

Assume that, UE enters the total DRX period (withprobability i∗(λ)c∗(λ)). In such a case,

E(BDRX ) =i∗(λ)c∗(λ)(λs + E(R))ρ

λ(1 − ρ).

To conclude

E(B) = E(Binact) + E(Bcl) + E(BDRX ) = ρE(W ).


Probabilities of UE activities

P(inact) = E(inact)

E(W )

P(cl) = E(cl)

E(W )

P(st) = E(st)

E(W )

P(sleep on short DRX ) = E(short DRX)−E(K )τ

E(W )

P(sleep on short DRX ) = E(long DRX)−E(M)τ

E(W )

P(receiving) = ρ1 = λ(1−β∗(θ))θ(β∗(θ)+p(1−β∗(θ)))

P(repair) = ρ2 = λ(1−β∗(θ))r(β∗(θ)+p(1−β∗(θ)))

P(listening) = (E(K )+E(N))τ

E(W )

Note that ρ = ρ1 + ρ2.


Power Saving Factor and Average Power Consumption

PSF and APC

PSF is the percentage of time during which UE is turnedoff and as a consequence does not consume power.Let Cinact , Ccl , Cst , CDS, CDL, CB, Clis power consumptionsof UE receiver at specific activities

Then

PSF = E(short DRX)−E(K )τ

E(W )+ E(long DRX)−E(M)τ

E(W )

= i∗(λ)c∗(λ)

E(W )[(1−(v∗

0 (λ))N)(v0−τ)

1−v∗

0 (λ)+

(v∗

0 (λ))N(v1−τ)

1−v∗

1 (λ)],

(4)

APC = E(inact)

E(W )Cinact +

E(cl)

E(W )Ccl +

E(st)

E(W )Cst

+E(short DRX)−E(K )τ

E(W )CDS + (E(K )+E(N))τ

E(W )Clis

+ρCB + E(long DRX)−E(M)τ

E(W )CDL.

(5)


Energy efficiency and cost function

Necessity for a metric that gives a fair trade off betweenenergy saving and delay for various sleep mode operations(see De Vuyst et al. [7]).

ηc = G − cD, (6)

where c is a delay penalty. The gain in energy and thecost on the delay by implementation of DRX LTE are givenby

G =APC(∗) − APC

APC(∗), D =

E(D)− E (∗)(D)

E(D).

where APC(∗), E (∗)(D) the average consumption and delayfor the model with single DRX period.

Our policy is efficient when ηc > 0.


Energy efficiency and cost function

Let t0, t1 the holding cost for each packet present in thesystem and the cost per unit power that UE consumes.The following function gives the total cost in the system.

T = t0E(X ) + t1APC.

An important metric of system’s reliability is the percentageof time the UE receiver is available, called UE availability

AV = ρ1 +E(inact)

E(W )+

E(cl)

E(W )+

E(st)

E(W )+

(E(K ) + E(N))τ

E(W )


Probability distributions are assumed to be exponential.Default values: b = c = 0.2 sec, r = 0.05 sec, p = 0.2,θ = 0.1, τ = 0.02 sec, t0 = 1, t1 = 2, CB = 300mW ,Crel = 30mW = CDS, CDL = 15mW , Cst = 45mW andClis = 105mW .

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Arrival rate

Pow

er s

avin

g fa

ctor

E(inactivity)=E(transmission)E(inactivity)=10E(transmission)E(inactivity)=60E(transmission)Lower limit of power saving factor

0 1 2 3 4 50

10

20

30

40

50

60

Arrival rate

Mea

n nu

mbe

r of

pac

kets

E(inactivity)=E(transmission)E(inactivity)=10E(transmission)E(inactivity)=60E(transmission)

Figure: PSF (a) and E(X ) (b) vs λ for N = 4, v0 = 2sec, v1 = 4sec.


0 0.5 1 1.5 2 2.5 3 3.5 40.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

Mean duration of short DRX cycle for E(inactivity)=0.2

Pow

er s

avin

g fa

ctor

E(listen)=0.1E(transmission)E(listen)=0.5E(transmission)E(listen)=0.8E(transmission)

2 4 6 8 10 12 14 16 18 200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Mean duration of long DRX cycle for E(inactivity)=0.2

Pow

er s

avin

g fa

ctor

E(listen)=0.2E(short DRX)E(listen)=0.5E(short DRX)E(listen)=0.8E(short DRX)

Figure: PSF vs v0 (v1 = 4sec) (a) and v1 (v0 = 2sec)(b) for N = 4,λ = 0.5, i = 0.2sec.


0 1 2 3 4 50

50

100

150

200

250

300

Arrival rate

Ave

rage

Pow

er C

onsu

mpt

ion

E(inactivity)=E(transmission)E(inactivity)=10E(transmission)E(inactivity)=60E(transmission)

1 2 3 4 5 6 7 8 9 1060

80

100

120

140

160

180

200

220

Mean duration of inactivity period

Ave

rage

Pow

er C

onsu

mpt

ion

Arrival rate=0.5Arrival rate=1Arrival rate=3

Figure: APC vs λ (a) and i (b) for N = 4, v1 = 4sec, v0 = 2sec.


0 5 10 15 2048

50

52

54

56

58

60

62

64

66

Fixed number of short DRX cycles N for arrival rate=0.5

Ave

rage

Pow

er C

onsu

mpt

ion

Mean duration of short DRX cycle=0.5Mean duration of short DRX cycle=1Mean duration of short DRX cycle=2

0 5 10 15 200.7

0.72

0.74

0.76

0.78

0.8

0.82


Pow

er s

avin

g fa

ctor

Mean duration short DRX=2Mean duration short DRX=1Mean duration short DRX=0.5

Figure: APC (a) and PSF (b) vs N for λ = 0.5, i = 0.2sec, v1 = 4sec.


0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mean duration of short DRX cycle for E(inactivity)=0.2

Ene

rgy

gain

Arrival rate=0.5Arrival rate=2

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Arrival rate

Ene

rgy

gain

Mean duration short DRX=2Mean duration short DRX=0.5Mean duration short DRX=0.25

Figure: Energy gain vs v0 (a) and λ (b) for N = 4, i = 0.2sec,v1 = 4sec.


0 0.5 1 1.5 2 2.5 3 3.5 4−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Mean duration of short DRX cycle for E(inactivity)=0.2, c=0.05

Eff

icie

ncy

Arrival rate=0.4Arrival rate=0.5Arrival rate=1

Figure: Efficiency η0.05 vs v0 for N = 4, v1 = 4sec, i = 0.2sec.


0 5 10 15 20 25 30 35 40−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12


Eff

icie

ncy

Mean duration short DRX cycle=2Mean duration short DRX cycle=0.5Mean duration short DRX cycle=0.25

Figure: Efficiency η0.05 vs N for λ = 0.5, v1 = 4sec, i = 0.2sec.


0 0.5 1 1.5 2 2.5 3 3.5 4−0.05

0

0.05

0.1

0.15

0.2

Mean duration of short DRX cycle for λ=0.5

Sle

ep

mo

de

eff

icie

ncy

Mean duration of long DRX cycle=4Mean duration of long DRX cycle=5Mean duration of long DRX cycle=10

Figure: Efficiency η0.05 vs v0 for λ = 0.5, N = 4, i = 0.2sec.


0 0.5 1 1.5 2 2.5 3 3.5 40.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

Mean duration of short DRX cycle for λ=0.5

Ava

ilab

ility

Mean inactivity period=0.2secMean inactivity period=0.1sec

Figure: Availability vs v0 for v1 = 4sec, λ = 0.5, N = 4.


0 0.5 1 1.5 2 2.5 3 3.595

100

105

110

115

120

125

130

Mean duration of short DRX cycle

Co

st

fun

ctio

n

N=2N=4N=8N=12

Figure: Cost function vs v0 for i = 0.2sec, λ = 0.5, v1 = 4sec.


Long Term Evolution (LTE): A Technical Overview. (2007)Technical White Paper, Motorola.

Baek, S. & Choi, B.D. (2011). Analysis of discontinuous reception(DRX) with both downlink and uplink packet arrivals in 3GPPLTE. Proceedings of QTNA 2011, Seoul, Korea. 8-16.

Bontou, C. & Illinge, E. (2009). DRX Mechanism for PowerSaving in LTE. IEEE Communications Magazine 47(6): 48-55.

Ross, S. (1996) Stochastic Processes, Wiley and Sons 2ndEdition.

Saffer, Z., Telek, M. (2010) Analysis of BMAP vacation queueand its application to IEEE 802.16e sleep mode. J. Ind. Manag.Opt. 6(3) 661-690.

Takacs, L. (1962). Introduction to the Theory of Queues, OxfordUniv. Press, New York.

S. De Vuyst, K. De Turck, D. Fiems, S. Wittevrongel, H. Bruneel.(2009) Delay versus Energy Consumption of the IEEE 802.16eSleep-Mode Mechanism. IEEE Trans. Wir. Com., 8(11)


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An unreliable vacation queueing model and its application ... · PDF fileIntroduction Model Mathematical analysis Performance metrics Numerical results References An unreliable vacation