An Approximate Queueing Model for Multi-rate

Multi-user MIMO systems

Boris Bellalta,Vanesa Daza, Miquel Oliver

Abstract

A queueing model for Multi-rate Multi-user MIMO systems is pre-sented. The model is built upon the assumption that the probabil-ity distribution of available destinations among the buered frames atthe Base Station (BS) is approximately the same as the probabilitydistribution of the trac arriving to the BS, this is, the amount oftrac directed to each MN with respect to the total trac load. Thisassumption leads to a simple, but accurate, queueing model for Multi-user MIMO systems that accounts for the impact of a nite number ofactive MNs in non-saturated conditions. The model is easily applicableto any Multi-user MIMO scenario given that the probability densityfunction of the post-processing SINR (Signal to Interference and NoiseRatio) for each MN is known.

Information about this paper:

This is an updated version of the paper published in IEEE Commu-nications Letters: Boris Bellalta, Vanesa Daza, Miquel Oliver; 'AnApproximate Queueing Model for Multi-rate Multi-user MIMO sys-tems'. IEEE Communications Letters, 2011.

The Matlab code for the described model can be found at:http://www.dtic.upf.edu/bbellalt/. For any comment about, pleasecontact Boris Bellalta (boris.bellalta@upf.edu).

This work has been partially supported by the Spanish Government under projectsTEC2008-0655 (Plan Nacional I+D), TEC2009-13000 (Plan Nacional I+D+i), CSD2008-00010 (Consolider-Ingenio Program) and by the Catalan Government (SGR2009#00617).The authors are with the Wireless Networks Research Group, Dept. de Tec-nologies de la Informacio i les Comunicacions, Universitat Pompeu Fabra, Spain(email:fboris.bellalta,vanesa.daza,miquel.oliverg@upf.edu).

1

r1

r2Scheduler

& Tx BF.

i

j

M antennas

ArrivalsFrame

Ideal Feedback Channel({i:h(1)..h(M)}, {SINR(1)...SINR(N)})

Figure 1: Specic scenario with M = 2 antennas serving frames to N MNs

1 Introduction

In a Multi-user MIMO system, a BS equipped with M antennas is able totransmit simultaneously up toM frames toM dierent single-antenna MNs,achieving a maximum spatial multiplexing gain of M [1]. However, this isonly possible if two conditions are satised: i) the BS has at leastM framesstored in its transmission queue and ii) these stored frames are destined toat least M dierent MNs. On the contrary, the number of stored frames orthe available destinations among them will limit the maximum number offrames that can be transmitted in parallel. Therefore, the queueing process(how the number of queued frames evolves with the time, which depends onboth the arrival and service processes) has to be considered in detail to reallyunderstand the performance that Multi-user MIMO systems can provide.

In this letter, a queueing model for Multi-rate Multi-user MIMO sys-tems in non-saturated conditions and with a nite number of active MNsis presented. From the physical layer, the model relies on the knowledge ofthe post-processing SINR distribution [2] at each MN, independently of theprecoding technique is used at the BS.

2 System Model and Assumptions

A BS with M antennas and a nite-buer of size K frames are considered(Figure 1). Frames of length equal to L bits (constant) directed to theN single-antenna MNs arrive to the BS following a Poisson process withaggregate rate , equally distributed among all active destinations.

A general precoding scheme is used at the BS, fed with the ChannelState Information (CSI) acquired at each MN and sent to the BS throughan ideal and instantaneous feedback channel. Based on those CSI values,a transmission rate, picked from a nite set of rates R, is used for thecommunication between the BS and the MNs in a frame-by-frame basis.

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A FIFO-based scheduling (frame selection) algorithm is considered. Itselects & 2 [1;min(q;M)] frames from those q 2 [1;K] frames stored in thequeue when a new transmission is scheduled, which happens just after theprevious one has nished or, if the queue has become empty, immediatelyafter the arrival of a new frame. Specically, the BS always schedules therst frame waiting for transmission and then, it selects sequentially up tomin(q;M) 1 frames, directed to not yet selected destinations. The groupof selected frames at each transmission is called a space-batch.

In Figure 2, a specic example of the queue behavior is shown forM = 2antennas, K = 3 frames and R = fr1; r2g rates. Observe that the (i + 1)-th space-batch comprises two frames, one transmitted at r1 and one at r2respectively, resulting in a space-batch duration of L=r1 seconds as r1 isthe minimum transmission rate in the (i + 1)-th space-batch. In general,let r = minfr1; : : : ; r&g be the lowest transmission rate assigned to oneof the frames included in a certain space-batch and p(r; &) its associatedprobability. The dependence with & in p(r; &) is included to account for theimpact that the number of spatially multiplexed streams can have on theassigned transmission rates.

q=2q=3 q=1 q=2 q=3

1/1/

ti1 i i+1

r2 r1

q=1 q=2 q=0q=0Tx Tx Tx TxNonTx

Figure 2: Temporal evolution of the BS's queue. The q values are the queuestate (number of frames in the queue) just after a frame arrival or a space-batch departure. Observe that r2 > r1.

3 The Queueing Model

AM=G[1;M ]=1=K batch-service queue [3] is used to model the BS (for furtherinformation, refer to [4] and references therein). The goal of the presentedmodel is to analytically obtain the system delay (average time that a frameremains in the BS) and the blocking probability (probability that a newarriving frame can not be stored in the queue as there is no free space init) for the considered Multi-user MIMO system. Two steps are followed to

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solve the presented model: i) the departure distribution, d, is computed byusing the discrete-time embedded Markov chain method and ii) the PASTA(Poisson Arrivals See Time Averages) property of the Poisson arrivals is ap-plied to nd the probability distribution at arbitrary times, s, as a functionof d.

3.1 Distribution of eligible frames in the queue (DEFQ)

As the trac load is uniformly distributed among all destinations, the prob-ability that an arriving frame is destined to a target MN is 1=N . Therefore,to compute the probability of selecting & 2 [1;min(q;M)] frames from thoseq frames stored in the queue, it is assumed that, given a randomly chosenframe from the BS queue, it is destined to a target MN with probability1=N too, that is, the same as its arrival probability.

Let Aq;&;N be the event that computes, when q frames are stored in thequeue, the possible choices of & frames directed to dierent destinations,among a set of N possible destinations. The probability of the event Aq;&;Ncan be computed as the quotient of favorable and total events. The numberof total events is N q, that is, the total number of dierent combinationsof q elements if each element can take N dierent values. To compute thenumber of favorable events, we should consider, given the presence of qframes, those & of them directed to dierent destinations. To do so, we rstx the & destinations, that is,

N&

. Then, for each one of these destinations,

all possible ordered partitions of & dierent elements in a queue with q frameshave to be considered. That is

P(1;:::;&)2q;& PR

q1;:::;& , where q;& is the

set dened as q;& = f(1; : : : ; &) 2 Z&+ j1 + : : : + & = qg and PRq1;:::;&denotes the permutation with repetition of q elements in sets of 1; : : : ; &elements. Thus, it derives in the following analytic formula:

p(Aq;&;N ) =

N&

Nq

X

(1;:::;&)2q;&PRq1;:::;& (1)

3.2 Distribution at departure epochs

The probability distribution at departure epochs, d, is computed solvingthe linear system d = dP, together with the normalization conditiond1T = 1. P is the probability transition matrix, where each i; j 2 [0;K]position, represents the probability pi;j to move from any state i, with i thenumber of frames in the queue at the moment that a new space-batch is

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scheduled, to any state j, with j the number of frames in the queue justafter the space-batch departure. Each pi;j is computed averaging all thedierent space-batch sizes and transmission rates that make that transitionpossible, that is

pi;j =

max(1;min(i;M))X&=1

p(Ai;&;N )X8r2R

p(r; &)pi;j(r; &) (2)

where p(Ai;&;N ) is computed from

p(Ai;&;N ) =

p(A1;&;N ); i = 0p(Ai;&;N ); i 1 (3)

as it takes into account the specic case in which the system is empty, p(r; &)is the probability that a space-batch involving & frames is transmitted at rater, as dened in Section 2, and pi;j(r; &) is the transition probability from anystate i to any state j given that a space-batch of & frames is transmitted atrate r.

From any state i 1 to any state j 2 [i &;K & 1], pi;j(r; &) iscomputed taking into account the probability of v = (ji)+& arrivals duringthe service time of the on-going space-batch, this is pi;j(r; &) =

(Lr)v

v! eL

r

where L=r is the space-batch service time and is the Poisson aggregatearrival rate. The transition probability from any state i 1 to the lastpossible state in which the queue can depart, j = K &, is computed as thecomplementary probability of not moving to any of the other states, this ispi;K&(r; &) = 1

PK&1j=i& pi;j(r; &). Finally, from the queue empty state,

i = 0, to any state j, the transition probability p0;j(r; &) is given by the sameprobability of departing in state q = 1, p0;j(r; &) = p1;j(r; &), as the systemremains inactive while the queue is empty.

3.3 Distribution at arbitrary times

The distribution at arbitrary times, s, is obtained using the PASTA prop-erty, which states that the probability to be at state u at any arbitrary timeis equal to the probability that a new arrival nds the queue at this state.Note that this is equivalent to say that the system has been in the u-th statefor 1= seconds on average in a given interdeparture epoch (Figure 2).

Additionally, from Figure 2, it can be observed also that the queueevolves between two macro-states, the non-transmitting and transmittingstates, with average duration E[Tt] and E[Tnt] respectively. The steady-stateprobability that a new arrival nds the queue in one of these two macro-states depends on the proportion of time that the system is in each one, that

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is t =E[Tt]

E[Tnt]+E[Tt]for the transmitting macro-state and nt = 1 t for

the non-transmitting one. E[Tt] depends on the transmission rate at whicha space-batch of size & frames is transmitted. Then, averaging over all

possible cases, E[Tt] =PK1

i=0 di

Pmax(1;min(i;M))&=1 p

(Ai;&;N )P

8r2R p(r; &)Lr .

E[Tnt] is computed taking into account that the time between two packetarrivals is in average 1= and the system is in the non-transmitting stateonly if previous departure has nished in the empty state (otherwise the non-transmitting macro-state has a duration equal to zero). Then, E[Tnt] =

1

d0 .

The probability that the system is in u-th state, u 2 [1;K 1], atany arbitrary time is shown in Equation 4. Note that, given a space-batchstarting at the i-th state, the transition probabilities that guarantee that anarrival has observed the queue at the u-th state before the system departs atstate j, depend on the size of the space-batch, the transmission rate r, andthe relative position of the u-th state with respect to the i-th state. This is,

su = t1

E[Tt]

1

uXi=0

di

0@max(1;min(i;M))X&=1

p(Ai;&;N )X8r2R

p(r; &)K&X

j=u+1πj(r; &)

1A(4)

The probability to be in the empty state at any arbitrary time is theprobability that a new arrival nds the system in the non-transmitting state,this is s0 = nt. Finally, the probability to be at the K-th state at anyarbitrary time is computed as sK = 1

PK1k=0

sk, given that

PKk=0

sk = 1.

4 Performance Results

Results from the queueing model are compared with the ones obtained bysimulation. Simulations are done in C++, carefully considering the de-scribed scenario. The considered parameters are: L = 1872 bits (constant),R = fr1; r2g, with r1 = 50 Kbps and r2 = 150 Kbps. Without loss of gen-erality, here it is assumed that the probability that r1 is the minimum rate

when & frames are scheduled is p(r1; &) =

&11

, with M a scaling

parameter. Then, the probability that a space-batch will be transmitted atr2 is p(r2; &) = 1 p(r1; &).

In Figure 3 the blocking probability (Pb = sK) is shown for K = 25

frames, two dierent number of antennas (M = 4 and M = 8) and twodierent number of MNs (N = 8 and N = 80) against the aggregate tracload in bits/second. In Figure 4 the average delay (queueing plus transmis-

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sion time, computed by applying Little's Law, E[D] =PK

k=0 ksk(1Pb) ) is shown

forM = 8 antennas, two queue sizes, K = 25 and K = 75 frames, a variablenumber of users and a xed trac load equal to 350 Kbps. In both cases, is set to 8 to penalize further the presence of M = 8 antennas (i.e. thechances to transmit at r2 with M = 4 antennas are higher).

The model shows a very good accuracy for the considered trac loads,number of antennas, number of active MNs and queue sizes. The pointswhere the queueing model shows the worst accuracy are those in which thenumber of users and the number of antennas are similar. In such situation,the DEFQ approximation is optimistic as the predicted chances to schedulelarger space-batches are higher than in the real system. Additionally, thereare some other conclusions that can be observed in the results. They can beexplained by the low number of eligible frames at each transmission, causedby both the consideration of non-saturated trac sources and the presenceof a nite buer. First, increasing the number of antennas does not mean aproportional gain on the eective system capacity (see Figure 3); second, asthe number of antennas grows, the performance gains are highly inuencedby the number of active MNs (see Figure 3 and observe that with M = 4antennas the same results are achieved for N = 8 and N = 80 MNs); andthird, increasing the number of active MNs allows the system to performmore eciently until a certain value where it stabilizes (see Figure 4). Fromthis value on, increasing the number of MNs does not provide any substantialgain.

5 Conclusions

A new queueing model for Multi-user MIMO systems in non-saturated con-ditions has been presented. The assumptions done, together with the appliedmodeling methodology, result in a simple, but accurate, model specially in-dicated to be considered for the design of cross-layer MAC/PHY protocolsfor such systems.

References

[1] David Gesbert, Marios Kountouris, Robert W. Heath Jr., Chan-ByoungChae, and Thomas Salzer. Shifting the MIMO paradigm. From single-user to multiuser communications. IEEE Signal Processing Magazine.Vol. 24, Page(s). 26-46, 2007.

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100 200 300 400 500 600106

105

104

103

102

101

100

Traffic Load (Kbps)

Bloc

king

Pro

babi

lity [K

25]

Model M=4,N=8Sim M=4,N=8Model M=4,N=80Sim M=4,N=80Model M=8,N=8Sim M=8,N=8Model M=8,N=80Sim M=8,N=80

Figure 3: Blocking Probabiltiy

100 101101

102

103

N (Active MNs)

Del

ay (m

secs

) [M8]

Model K=25Sim K=25Model K=75Sim K=75

Figure 4: Delay

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[2] Chiung-Jang Chen, and Li-Chun Wang; Performance Analysis ofScheduling in Multiuser MIMO Systems with Zero-Forcing Receivers.IEEE Journal on Selected Areas in Communications, Vol.25, Number:7,Page(s): 1435-1445, 2007.

[3] Donald Gross, and Carl M. Harris; Fundamentals of Queueing Systems,3er Ed.. John Wiley & Sons, 1998.

[4] Boris Bellalta, and Miquel Oliver; A Space-Time Batch-service QueuingModel for Multi-user MIMO communication systems. In In Proceedingsof the 12-th ACM International Conference on Modeling, Analysis andSimulation of Wireless and Mobile Systems (MSWIM), Tenerife, Spain,2009.

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