The Stability Region of the Two-User Broadcast Channel

0.5 MB·English

by  Nikolaos Pappas

The Stability Region of the Two-User Broadcast
Channel
Nikolaos Pappas*, Marios Kountouris†,
*Department of Science and Technology, Linko¨ping University,
Campus Norrko¨ping, 60 174, Sweden
†Mathematical and Algorithmic Sciences Lab, France Research Center, Huawei Technologies Co. Ltd.
Boulogne-Billancourt, 92100, France
Emails: [email protected], [email protected]
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1 Abstract—In this paper, we characterize the stability region channel. The work in [6] provided a partial characterization
0 ofthetwo-userbroadcastchannel.First,weobtainthestability of the capacity region of the two-user Gaussian fading
2 region in the general case. Second, we consider the particular broadcast channel. Caire and Shamai in [7] investigated the
case where each receiver treats the interfering signal as noise,
b achievablethroughputofamulti-antennaGaussianbroadcast
as well as the case in which the packets are transmitted using
e superpositioncodingandsuccessivedecodingisemployedatthe channel. In [8], scheduling policies in a broadcast system
F
strong receiver. were considered and general conditions covering a class of

3 throughput optimal scheduling policies were obtained. In
2 I. INTRODUCTION [9], the authors characterized the stability regions of two-
AfundamentalquestiontowhichInformationTheoryaims userGaussianfadingmultipleaccessandbroadcastnetworks
]
T to provide an answer is how to maximize the use of a with centralized scheduling under the assumption of infinite
I communication channel between transmitters and receivers. backloggedusers.In[10],thecapacityregionofthetwo-user
s. In other words, its major objective is to characterize the broadcast erasure channel was characterized and algorithms
c maximum achievable rate of information that can be reliably basedonlinearnetworkcodingandtheirstabilityregionwere
[
transmitted over a communication channel, which is called also provided.

3 the channel capacity. In contrast to point-to-point channels, Superposition Coding (SC) [3] is one of the fundamental
v if the channel is shared among multiple nodes (multiuser building blocks in network information theory. The ob-
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channel), the goal is to find the capacity region, i.e. the jective of SC is to simultaneously transmit two messages
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2 set of all simultaneously achievable rates. One of the main by encoding them into a single signal in two layers. The
6 assumptions in the information-theoretic formulation of the receiver with the “better” (less noisy) channel, also named
0 capacity region is that the maximum achievable rate is stronger receiver, can recover the signal on both layers by
1. obtained under infinitely backlogged users. However, the applying successive interference cancelation, while the other
0 burstynatureofthesourcesincommunicationnetworksgave (weaker or “worse”) can decode the message on the coarse
5 rise to the development of a different concept of “capacity layer treating the message on the fine layer (interference) as
1 region”, which is the maximum stable throughput region noise. In [11], SC with conventional frequency division in
:
v or the stability region [1]. Understanding the relationship a Poisson field of interferers was analyzed. Furthermore, in
Xi between the information-theoretic capacity region and the [12], the authors provided a software-radio based design and
stability region has received considerable attention in re- implementationofSC.TheirresultsshowthatSCcanprovide
r
a cent years and some progress has been made primarily for substantial spectral efficiency gains compared to orthogonal
multiple access channels. Interestingly, the aforementioned schemes, such as time division multiplexing. The stability
regions (capacity and stability) are not in general identical region of the two-user interference channel was derived in
and general conditions under which they coincide are known [13], where the case of successive interference cancelation
only in very few cases [2]. was also considered.
In this work, we consider the two-user broadcast channel Inthispaper,thestabilityregionofthetwo-userbroadcast
[3], which models the simultaneous communication of in- channel is obtained. We first provide the stability region for
formation (different messages) from one source to multiple the general case as a function of success probabilities and
destinations. Marton in [4] derived an inner bound, which afterwardswespecializeourstudyconsideringtwoparticular
is the best known achievable information-theoretic capacity cases. The first case is when both receivers treat interference
region for a general discrete memoryless broadcast channel. as noise. The second case is when superposition coding
Fayolle et al. [5] provided a theoretical treatment of some is employed and the user experiencing better channel uses
basic problems related to the packet switching broadcast a successive decoding scheme. Two simple transmit power
allocation schemes are considered in the latter case: i) the
ThisworkhasbeensupportedbythePeopleProgramme(MarieCurieAc-
assigned power remains fixed, and ii) the transmit power is
tions)oftheEuropeanUnion’sSeventhFrameworkProgrammeFP7/2007-
2013/underREAgrantagreementno.[612361]–SOrBet. adapted to the state of the queues.
λ1 λ2 where nt is the additive white Gaussian noise at timeslot t
i
with zero mean and unit variance. The channel gain from
Queue 1 Queue 2 athnedttrhaensmtraitntsemr ittotedDisigantatlimisexinti.staAntbtlocisk dfeandointegdcbhyannhetil,
model with Rayleigh fading is considered here, i.e. the
S
fading coefficients ht remain constant during one timeslot,
i
butchangeindependentlyfromonetimeslottoanother.Inthe
h1 h2 transmissionphase(downlink),thetransmitterassignspower
P for messages (packets) from queue i.
i
D1 D2 The event Di/i is defined as the probability that the
uncoded received signal-to-noise ratio (SNR) is above a
certain threshold γ , i.e. = γ SNR . The distance
Fig. 1: The two-user broadcast channel with bursty arrivals. i Di/i { i ≤ i}
betweenthetransmitterandD isdenotedbyd andαisthe
i i
pathloss exponent. The SNR threshold for receiver i is γ .
i
Then SNR (cid:44) h 2d−αP assuming a physical layer model.
II. SYSTEMMODEL i | i| i i
The probability that the link between the transmitter and D
i
We consider a two-user broadcast channel, as depicted is not in outage when only the i-th queue is non-empty is
in Fig.1, in which a single transmitter having two different given by (Ch. 5.4 in [14])
queues intends to communicate with two receivers. The first
γ dα
(resp.second)queuecontainsthepackets(messages)thatare Pr =Pr SNR γ =exp i i . (4)
destined to receiver D1 (resp. D2). Time is assumed to be (cid:16)Di/i(cid:17) { i ≥ i} (cid:18)− Pi (cid:19)
slotted,thepacketarrivalprocessesatthefirstandthesecond Note that this is an approximation on the success probability
queue are assumed to be independent and stationary with under specific assumptions on the underlying physical layer
mean rates λ1 and λ2 in packets per slot, respectively. Both model and is done in order to relate the success probabilities
queues have infinite capacity to store incoming packets and withaphysicallayerandchannelmodel.Actually,theabove
Qi denotes the size in number of packets of the i-th queue. expression on the success probability comes from the rates
Thesourcetransmitspacketsinatimeslotifatleastoneofits forarbitrarilyreliablecommunication,whichimpliesthatthe
queue is not empty. The transmission of one packet requires channel uses go to infinity. This (asymptotic) expression is
one timeslot and we assume that receive acknowledgements anapproximationoftheinstantaneousratewheninformation
(ACKs) are instantaneous and error-free. istransmittedinpackets.Nevertheless,themainresultofthis
If only the i-th queue at the source is non-empty during paper,i.e.thestabilityregionderivedinthefollowingsection,
a certain timeslot, then the transmitter sends information to is general as it is expressed in terms of success probabilities,
the i-th receiver only. When both queues have packets, the which can in turn take on different expressions depending
source transmits a packet that contains the messages of both on the adopted physical channel or the information-theoretic
receivers, whereas whenever both queues at the source are model.
empty, the transmitter remains silent. Whenboth queuesat thesource arenon-emptyat timeslot
Let Di/T denote the event that Di is able to decode the t then the source transmits the signal xt =xt1+xt2 where xti
packet transmitted from the i-th queue of the transmitter is the signal for the user D . Then, the signal y received at
i i
given a set of non-empty queues denoted by T, e.g. D1/1,2 user Di at a timeslot t is given by yit =htixt+nti,i=1,2,
denotes the event that the first receiver can decode the where nt is the additive white Gaussian noise at timeslot
i
packet from the first queue when both queues are not empty t with zero mean and unit variance. The channel gain
( = 1,2 ). It is evident that Pr Pr . from the transmitter to D is denoted by ht at instant t
T { } D1/1 ≥ D1/1,2 i i
The average service rate seen by the first queue is and a Rayleigh block fading model is considered. In the
(cid:16) (cid:17) (cid:16) (cid:17)
transmissionphase(downlink),thetransmitterassignspower
(cid:16) (cid:17) (cid:16) (cid:17)
µ1=Pr(Q2>0)Pr D1/1,2 +Pr(Q2=0)Pr D1/1 . (1) Pi for messages (packets) of queue i with P1 +P2 = P.
We assume that each receiver D knows each channel ht
i i
(perfect CSIR) and that the transmitter has perfect channel
Respectively, the average service rate of the second queue
stateinformation(CSIT),i.e.itknowsht, i.Eachreceiveri
is i ∀
decodes separately its message using the received signal y .
i
(cid:16) (cid:17) (cid:16) (cid:17) Thesuccessprobabilitiesinthecasethatbothqueuesarenon-
µ2=Pr(Q1>0)Pr D2/1,2 +Pr(Q1=0)Pr D2/2 . (2)
emptydependontheinterferencehandlingtechniqueandfor
thatwestudycertaindifferentcasesinthefollowingsections.
Ifapacketfromthei-thqueuefailstoreachD ,itremains
i
A. Stability Criteria
in queue i and is retransmitted in the next timeslot.
The signal y received at user D at a timeslot t is given We use the following definition of queue stability [15]:
i i
by
Definition 1. Denote by Qt the length of queue i at the
i
yt =htxt+nt (3) beginning of time slot t. The queue is said to be stable if
i i i i
lim Pr[Qt < x] = F(x) and lim F(x) = 1. If �
t→∞ i x→∞ 2
lim lim infPr[Qt <x]=1,thequeueissubstable.
If ax→qu∞eue ist→st∞able, then iit is also substable. If a queue is Pr D1/1,2 +Pr D2/1,2 >1
not substable, then we say it is unstable. Pr D2/2 Pr�D1/1 � Pr�D2/2 �(<1
� � � �
� �
Loynes’ theorem [16] states that if the arrival and service
processes of a queue are strictly jointly stationary and the
average arrival rate is less than the average service rate, then
the queue is stable. If the average arrival rate is greater than Pr
D2/1,2
the average service rate, then the queue is unstable and the � �
value of Qt approaches infinity almost surely. The stability
i
regionofthesystemisdefinedasthesetofarrivalratevectors
λ=(λ ,λ ) for which the queues in the system are stable.
1 2
�
Pr D1/1,2 Pr D1/1 1
III. THESTABILITYREGION–THEGENERALCASE � � � �
Fig. 2: The stability region for the two-user broadcast
Inthissection,weprovidethestabilityregionasafunction channel in the general case.
of the success probabilities in the general case without
considering specific interference handling techniques.
Theaverageserviceratesofthefirstandsecondqueueare B. Second Dominant System: the second queue transmits
given by (1) and (2), respectively. Since the average service dummy packets
rate of each queue depends on the queue size of the other In the second dominant system, when the second queue
queue, it cannot be computed directly. Therefore, we apply empties then the source transmits a dummy packet for the
the stochastic dominance technique [1], i.e. we construct D while the first queue behaves in the same way as in the
2
hypothetical dominant systems, in which one of the sources original system. In this dominant system, the second queue
transmits dummy packets when its packet queue is empty, never empties, so the service rate for the first queue is
while the other transmits according to its traffic.
µ =Pr . (9)
1 1/1,2
D
A. First Dominant System: the first queue transmits dummy (cid:16) (cid:17)
The first queue is stable if and only if λ < µ . The
packets 1 1
probability that Q is empty is
1
Inthefirstdominantsystem,whenthefirstqueueempties,
λ
then the source transmits a dummy packet for the D1, Pr(Q1 =0)=1− 1 . (10)
Pr
while the second queue behaves in the same way as in the 1/1,2
D
original system. All other assumptions remain unaltered in (cid:16) (cid:17)
The service rate of the second queue, after substituting
thedominantsystem.Thus,inthisdominantsystem,thefirst
(10) into (2) is
queue never empties, thus the service rate for the second
queue is given by µ2 =Pr D2/1,2 . µ =Pr Pr D2/2 −Pr D2/1,2 λ . (11)
Then, we can obtain sta(cid:16)bility co(cid:17)nditions for the second 2 D2/2 − (cid:16) Pr(cid:17) (cid:16) (cid:17) 1
queue by applying Loyne’s criterion [16]. The queue at the (cid:16) (cid:17) D1/1,2
(cid:16) (cid:17)
second source is stable if and only if λ < µ , thus λ < Thestabilityregion obtainedfromtheseconddominant
2 2 2 2
R
Pr . Then we can obtain the probability that the system is given in (8).
2/1,2
D
second queue is empty by applying Little’s theorem and is The stability region of the system is given by =
(cid:16) (cid:17) R
given by 1 2, where 1 and 2 are given by (7) and (8)
R R R R
respectively and is depicted by Fig. 2.
Pr(Q =0)=1 λ2 . (5) A(cid:83)n important observation made in [1] is that the stability
2
− conditions obtained by the stochastic dominance technique
Pr
2/1,2
D are not only sufficient but also necessary conditions for
(cid:16) (cid:17)
the stability of the original system. The indistinguishability
After replacing (5) into (1), we obtain that the service rate
argument [1] applies to our problem as well. Based on the
for the first queue in the first dominant system is
constructionofthedominantsystem,itiseasytoseethatthe
queuesofthedominantsystemarealwayslargerinsizethan
Pr Pr
1/1 1/1,2
µ =Pr D − D λ . (6) thoseoftheoriginalsystem,providedtheyarebothinitialized
1 D1/1 − (cid:16) (cid:17) (cid:16) (cid:17) 2
Pr to the same value. Therefore, given λ <µ , if for some λ ,
(cid:16) (cid:17) D2/1,2 2 2 1
the queue at S is stable in the dominant system, then the
(cid:16) (cid:17) 1
The first queue is stable if and only if λ < µ . The corresponding queue in the original system must be stable.
1 1
stability region obtained from the first dominant system Conversely,ifforsomeλ inthedominantsystem,thequeue
1 1
R
is given in (7). atnodeS saturates,thenitwillnottransmitdummypackets,
1
Pr Pr
λ 1/1 1/1,2
= (λ ,λ ): 1 + D − D λ <1,λ <Pr (7)
R1  1 2 (cid:16) (cid:17) (cid:16) (cid:17) 2 2 D2/1,2 
 Pr D1/1 Pr D1/1 Pr D2/1,2 (cid:16) (cid:17)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
 
Pr Pr
λ 2/2 2/1,2
= (λ ,λ ): 2 + D − D λ <1,λ <Pr (8)
R2  1 2 (cid:16) (cid:17) (cid:16) (cid:17) 1 1 D1/1,2 
 Pr D2/2 Pr D2/2 Pr D1/1,2 (cid:16) (cid:17)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
 
andaslongasS hasapackettotransmit,thebehaviorofthe Thus,
1
dominant system is identical to that of the original system
Pr =
because dummy packet transmissions are eliminated as we D2/1,2
approach the stability boundary. Therefore, the original and (cid:16) (cid:17)
∞ γ
the dominant system are indistinguishable at the boundary 1 F x 2 f (x)dx.
points. (cid:90)0  − |h2|2(cid:32) ≥ d−2α(P2−γ2(P −P2))(cid:33) |h2|2
The obtained stability region for the two-user broadcast   (14)
channel in the general case has the same expression with the Notethatf|h2|2(x)=exp(−x)andF|h2|2(x)=1−exp(−x).
stabilityregionofthetwo-userinterferencechannelobtained To summarize, the success probability for the second user
in [13]. when both queues at the source are non-empty is given by
(15), where 1 is the indicator function.
IV. TREATINGINTERFERENCEASNOISE {·}
Thus if P > γ P and P > γ P then Pr =
In this section, we consider the case where the users treat 2 2 1 1 1 2 D1/1,2
the interfering signal as noise. When the i-th queue is empty exp γ1dα1 andPr =exp γ(cid:16)2dα2 a(cid:17)fter
−P1−γ1P2 D2/1,2 −P2−γ2P1
at the source, while the j-th queue is not, then the success repla(cid:16)cing in (7)(cid:17)and (8) w(cid:16)e obtai(cid:17)n the sta(cid:16)bility region(cid:17) =
R
probabilityforthei-thuserisgivenby(4).Whenbothqueues .
1 2
R R
are non-empty then the transmitted signal at timeslot t from Similarly we can obtain the region for the other cases.
the source to the receivers is denoted by xt = xt + xt. (cid:83)
The received signal yt by the user D is yt = htx1t +n2t. V. SUCCESSIVEDECODING
i i i i i
The event denotes that user D is able to decode its Inthissectionweconsiderthecasewherethechannelfrom
i/i,j i
intended paDcket. This is feasible when the received SINR is thetransmittertoD1 isbetterthanthattoD2;i.e. h1 > h2 .
| | | |
above a threshold γ and is expressed by When only one queue at the transmitter is non-empty, then
i
the procedure of a successful transmission is described in
P h 2d−α Section II. When both queues at the source are non-empty,
= i| i| i γ . (12)
Di/i,j (cid:40)1+Pj|hi|2d−i α ≥ i(cid:41) theWperorceefderurteoothfedetwcoodpinagckaeptsacukseetdbiynaarseicnegilveersuispearspfoosliltoiwons-.
Thesuccessprobabilityofthesecondusercanbeobtained
based transmission as two levels (layers). The packet in-
similarlytothefirst.ThetransmissionfromthesourcetoD
2 tended for the weaker receiver (i.e. D ) is referred to as
2
is successful when
the first level. We refer to the other level as the second
P2|h2|2d−2α γ γ h 2d−α(P γ P ) level. A transmitter using superposition coding splits the
1+P h 2d−α ≥ 2 ⇐⇒ 2 ≤| 2| 2 2− 2 1 available transmission power between the two level, selects
1| 2| 2
γ h 2d−α(P γ (P P )). thetransmissionrateforeachofthelevels,thenencodesand
⇐⇒ 2 ≤| 2| 2 2− 2 − 2 modulates each of the packets separately at the selected rate.
Note that P1+P2 =P. Thus, if P2 γ2(P P2)<0 then, The modulated symbols are scaled appropriately to match
− −
the success probability is zero because the initial inequality thechosenpowersplitandsummedtoobtainthetransmitted
is not feasible. Thus, if P2 > 1+γ2γ2P then signal. More details about implementation of superposition
γ coding at the medium access layer can be found in [17].
h 2 2 .
| 2| ≥ d−2α(P2−γ2(P −P2)) noAiset tahnedrdeceeciovdeerssiitdse,daDta2 ftrroematsyth.eRmeceesisvaegreDof, Dw1hicahs
Assuming Rayleigh block fading, we have h 2 exp(1) i 1
| 2| ∼ has a better channel, performs successive decoding, i.e. it
and the success probability can be expressed as
decodes first the message of D , then it subtracts it from the
2
γ received signal, and afterwards decodes its message with a
Pr =Pr h 2 2 =
(cid:16)D2/1,2(cid:17) (cid:34)| 2| ≥ d−2α(P2−γ2(P −P2))(cid:35) ssiunpgelrep-oussietriodneccooddeirn.gN, othteetdheactoindinthgeobrrdoeardcisastdicfhfearnennetlfwroitmh
∞ γ2 SIC in the interference channel, in which the signal with the
= Pr x f (x)dx.
(cid:90)0 (cid:34) ≥ d−2α(P2−γ2(P −P2))(cid:35) |h2|2 strongest channel is decoded first. The successive decoding
(13) is feasible at the first receiver if
γ γ dα γ dα
Pr =1 P > 2 P exp 2 2 =1 P >γ P exp 2 2 (15)
D2/1,2 2 1+γ −(P γ (P P )) { 2 2 1} −P γ P
(cid:16) (cid:17) (cid:26) 2 (cid:27) (cid:18) 2− 2 − 2 (cid:19) (cid:18) 2− 2 1(cid:19)
�
2
P h 2d−α
2| 1| 1 γ , P h 2d−α γ . (16)
(cid:40)1+P1|h1|2d−1α ≥ 2 1| 1| 1 ≥ 1(cid:41) Pr D2/2
D is able to decode its intended packet if and only if
2 � �
the received signal-to-interference-plus-noise ratio (SINR) is
greater than γ .
2 Pr
The success probability seen by the first user, D1 when D2/1,2
both queues are non-empty is given by (17). The proof is � �
omitted due to space limitations.
The probability that the link between the transmitter and
�
D2 isnotinoutagewhenbothqueuesarenon-emptyisgiven Pr 1
1/1
by (15) which is obtained in the previous section. D
In the remainder, we consider two simple schemes regard- Fig. 3: The stability region for fixed tra�nsmit�powers when
ing the transmission power for each receiver’s packets. The P >P γ2(1+γ1) and D applies successive decoding and
first scheme is the case where we have fixed transmit power 2 1 γ1D treats i1nterference as noise.
2
P for the i-th receiver, such that P +P =P. The second
i 1 2
scheme comes naturally whenever a user is inactive, i.e. has
nopacketstoreceive.Weconsiderthatthetransmitteradapts 2) Thecasewhereγ P <P P γ2(1+γ1): Inthiscase
the powerconsidering thequeue stateof each receiver,i.e. if 2 1 2 ≤ 1 γ1
clearly Pr =Pr . We obtained that
the queue Qi is empty, then all power P is allocated to the D1/1,2 (cid:54) D1/1
j-th queue, (i=j). (cid:16) (cid:17) (cid:16) (cid:17) γ dα
(cid:54) Pr =exp 2 1 . (21)
A. Fixed Power Scheme D1/1,2 −P γ P
(cid:16) (cid:17) (cid:18) 2− 2 1(cid:19)
We assume here that the transmitter assigns fixed power
In this case the queues are coupled so, we have to use
P (resp. P ) at the D (resp. D ) on every timeslot.
1 2 1 2 the results obtained in Section III derived by the stochastic
1) The case where P > P γ2(1+γ1): The service rate
2 1 γ1 dominance technique. After replacing (21) and (15) into (7)
seen by the first queue is given by (1). Since constant
and (8) we obtain the stability region.
transmitting power P is used and D has better channel
1 1
than D , from (17) we have that Pr =Pr . B. Variable Power Scheme based on Queue State
2 1/1,2 1/1
D D
Thus, we have (cid:16) (cid:17) (cid:16) (cid:17) In this part, we consider a simple adaptive scheme re-
gardingthepowerallocationforeachreceiver’spackets.The
µ =Pr . (18)
1 1/1
D power allocation is performed as follows: when both queues
From Loyne’s criterion fo(cid:16)r stabi(cid:17)lity [16], the first queue is are not empty, the transmit power for the first and second
stableifandonlyifλ1 <µ1.FromLittle’stheorem(Ch.3.2 queue is P1 and P2, respectively, satisfying P1 +P2 = P.
in [18]), we have that However, when the queue of i-th receiver is empty, the total
λ transmit power P is used for transmitting the packets from
Pr(Q >0)= 1 . (19)
1 for the j-th (where j =i) receiver.
Pr 1/1 (cid:54)
D Theaverageserviceratesofthefirstandthesecondqueue,
Theservicerateforthesecond(cid:16)queue(cid:17)isgivenby(2).After µ ,andµ aregivenby(1)and(2)respectively.Thesuccess
1 2
substituting (19) into (2) we obtain
probabilities Pr for i=1,2 are given by
i/i
D
µ2 =Pr D2/2 + Pr(cid:16)D2/P1,r2(cid:17)−Pr(cid:16)D2/2(cid:17)λ1. (20) Pr(cid:16)Di/i(cid:17)=exp −γiPdαi , (25)
(cid:16) (cid:17) D1/1 (cid:16) (cid:17) (cid:18) (cid:19)
From Loyne’s criterion we hav(cid:16)e that(cid:17)the second queue is sincewhenaqueueisempty,thetransmitterassignsallpower
stable if and only if λ < µ . The stability region for the totheotherqueue,andcanbeobtainedfrom(4).Thesuccess
2 2
degraded broadcast channel is given by (22) is depicted by probability Pr is given by (17).
1/1,2
D
Fig. 3.
In the abov(cid:16)e schem(cid:17)e, it is evident that Pr =
Recall that the success probability Pr 2/1,2 is given D1/1 (cid:54)
by(15). Notethat inthiscase wedonot facDetheproblem of Pr , and as a result, there is coupling b(cid:16)etween(cid:17)the
(cid:16) (cid:17) 1/1,2
D
coupled queues as mentioned in the general case described queues.Thuswecanusedirectlythestabilityregionobtained
(cid:16) (cid:17)
in Section III. in Section III by replacing the success probabilities.
γ (1+γ ) γ dα γ (1+γ ) γ dα
Pr =1 γ P <P P 2 1 exp 2 1 +1 P >P 2 1 exp 1 1 (17)
D1/1,2 2 1 2 ≤ 1 γ −P γ P 2 1 γ − P
(cid:16) (cid:17) (cid:26) 1 (cid:27) (cid:18) 2− 2 1(cid:19) (cid:26) 1 (cid:27) (cid:18) 1 (cid:19)
Pr Pr
λ 2/2 2/1,2
= (λ ,λ ): 2 + D − D λ <1,λ <Pr (22)
R  1 2 (cid:16) (cid:17) (cid:16) (cid:17) 1 1 D1/1 
 Pr D2/2 Pr D1/1 Pr D2/2 (cid:16) (cid:17)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
 
= (λ ,λ ): λ1 + exp −γ1Pdα1 −exp −γP1d1α1 λ <1,λ <exp γ2dα2 (23)
R1  1 2 exp −γ1Pdα1 exp −(cid:16)γ1Pdα1 e(cid:17)xp −(1(cid:16)+γγ22)dPα22−(cid:17)γ2 2 2 (cid:18)−(1+γ2)P2−γ2(cid:19)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
 
= (λ ,λ ): λ2 + exp −γ2Pdα2 −exp −(1+γγ22)dPα22−γ2 λ <1,λ <exp γ1dα1 (24)
R2  1 2 exp −γ2Pdα2 (cid:16)exp −(cid:17)γ2Pdα2 ex(cid:16)p −γP1d1α1 (cid:17) 2 1 (cid:18)− P1 (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
 
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