On the Relationship Between Queuing Delay and Spatial Degrees of Freedom in a MIMO Multiple Access Channel

Transcription

1 On the Reationship Between Queuing Deay and Spatia Degrees of Freedom in a IO utipe Access Channe Sriram N. Kizhakkemadam, Dinesh Rajan, andyam Srinath Dept. of Eectrica Engineering Southern ethodist University Daas, TX Emai: {skizhakk, rajand, mds}@engr.smu.edu Abstract In this paper, we study the reationship between queuing deay for a random packet arriva process and physica ayer parameters in a mutipe access channe with mutipe antennas at the input and output. Our main contribution is the derivation of a simpe, anaytica approximation for the average deay that ceary indicates the effect of number of transmit and receive antennas, transmission power, the packet arriva rate and the desired reiabiity. Comparison with numerica anaysis indicates that the proposed anaytica approximation of the deay is accurate for medium and arge SNRs. I. INTRODUCTION The information age has seen a profusion of wireess mobie devices that has aowed users to communicate using voice, data and even mutimedia. A proper characterization of the achievabe performance is required so that users can then be given various service guarantees against a price differentia. Some of the parameters of the Quaity of service incude data rate of transmission, the fideity of reproducing the information at the receiver(distortion), reiabiity of the data ink (Probabiity of error), the deay in the avaiabiity of the information and the power required for information transmission. Unti the eary 9 s, majority of the research in both network information theory and networking impicity assumed the Open Systems Interconnection Basic Reference ode (OSI ode for short) of a network. The OSI mode is an ayered, abstract description for communications and protoco design. In this mode, a networking system is divided into various ayers. Within each ayer, one or more entities impement its functionaity. Each entity interacts directy ony with the ayer immediatey beneath it, and provides faciities for use by the ayer above it. The communication between entities in different ayers and possiby different hosts is through protocos ]. The study of the impact of physica ayer parameters on the network ayer has gained importance of ate in the fied of cross-ayer optimization. The deay at the physica ayer was typicay considered to be the time required to transmit and decode a codeword with or without ARQ (Automatic Repeat Request). The effect of random arriva of messages on the physica ayer was not given importance. For muti-user systems, Teatar and Gaager aid the foundation of an eegant formuation to determine the queuing deay in a singe antenna mutipe access channe (AC) in their semina paper 3]. This paper effectivey dispensed of with the independence notion of the different ayers of the OSI mode. A processor sharing queue was used to mode the queuing in a AC. In this mode, the service process is constant with time. The case where the service rate fuctuates was studied in 4]. Detais of the Teatar and Gaager mode wi be discussed ater. Recenty more efforts on understanding the queuing deay for IO systems have been undertaken 5], 6]. In 6], we studied the queuing deay of a IO AC with singe user decoding (SUD). Our resuts from numerica anaysis showed that depending on the number of receive antennas, increasing the number of transmit antennas can either increase or decrease the deay. Is this a imitation of singe user decoding? Is there a simpe reationship between queuing deay and the spatia parameters? otivated by these questions, we study the queuing deay with joint decoding (JD) in this paper. JD heps us give a ower bound on the minimum achievabe deay. In SUD, the received signa from other users is treated as noise. Instead, in JD, we use apriori knowedge of the set of a possibe codeword from a users at the receiver for say, successive interference canceation 7]. Our main contribution is the derivation of a cosed form expression for the queuing deay with joint decoding that accuratey matches with numerica evauation in the imit of arge SNR. The anaytica expression for deay ceary shows the dependence on system parameters ike SNR, number of antennas, probabiity of error, message size and arriva rate of packets. We observe that the deay is inversey proportiona to the minimum of number of transmit and receive antennas. The system mode we espouse is in cose concordance with that given in 3]. Notaby, the mode chosen in 3] bridges modes in information theory and queuing theory with a sense of engineering intuitive appea. After describing the system mode in section, we summarize the reationship between queuing deay and random coding error exponent in section 3. In section 4, we derive an approximate anaytica reationship between the deay and number of transmit and receive antennas that is vaid in the high SNR regime. We investigate the accuracy of the approximations in section 5 and concude in section 6. II. SYSTE ODEL Consider a symmetric IO AC with mutipe users each of which has transmit antennas. A the users transmit

2 their information to a singe receiver with N receive antennas. essages are generated according to a Poisson process with rate. Each new message is considered to represent a virtua user; each message is of size og K nats. Thus, we have in effect possiby infinite number of transmitters. This abstraction of a user transmitting a singe message is usefu in modeing the system as a processor sharing queue with the queue at the transmitter being effectivey modeed as a queue at the receiver for anaytica tractabiity 3]. Due to this virtua user/transmitter mode, we use the terms user and transmitter interchangeaby. We denote by, the random variabe associated with the number of users in the system at time t. Each user encodes its message into an infinite ength codeword for transmission. However, the entire codeword is not transmitted. After successfuy decoding the transmitted codeword, the receiver sends a signa to the appropriate user to cease transmission. In the imit of arge codeword engths, this bit of feedback is negigibe. The received signa y C N at the N receive antennas at time t depends on the transmitted signa according to, SNR y t H ix i,t + z t, () i where x i,t, H i and z i are the normaized input, channe and noise random variabes. The normaized input vector of the i th user, x i has entries distributed according to a compex Gaussian distribution with mean and unit variance. The rea and imaginary entries of the channe matrix H i C N are Gaussian distributed with mean and unit variance. The additive uncorreated white Gaussian noise, z is distributed according to CN(, ). The bandwidth at the receiver is W Hz. The factor of SNR takes into account arbitrary channe gains and non-unit receiver noise power. We assume that the transmitter has no knowedge of the channe and hence the transmitter of each user aocates the tota avaiabe transmit power equay to a the transmit antennas. However, the receiver has perfect channe state information (CSIR). We aso denote by H u {H,...,H u }. The symmetric IO AC with same number of transmit antennas and power constraints for a users has been chosen for anaytica convenience. Ony natura ogarithms are considered in this paper. In this paper, we consider the channe to be fast fading where the channe codeword is coded across mutipe coherence time intervas (fading bocks). We now give a summary description of modeing the queuing deay from an information theoretic perspective in the foowing section. III. QUEUING DELAY The theory of error exponents is used in 3] to reate queuing and physica ayer parameters. The error exponent term E o is reated to the service rate whie the specification on a desired probabiity of error and message size is reated to the demand. Based on a symmetric queue processor sharing mode as given in 8], a numerica evauation of the end-to-end deay was obtained in 3]. Specificay, in 3], the random coding bound on the probabiity of error is used to draw a parae between the random coding exponent term E o and the service rate for the mutipe access queue. From definition of random coding error exponent for continuous channes 9], the random coding bound is given by P e exp max ρ ρ og K E o (ρ, H u,q x ) () The error exponent term E o is, +ρ E o max og q x p (y, H u x) dx] /(+ρ) dy q x H u y x Since the channe is known at the receiver, E o simpifies to, +ρ E o max og E Hu q x p (y x, H u ) dx] q x y /(+ρ) dy x (3) where, the maximum is over a input distributions q x of the codeword and p (y x, H u ) πi N exp y H i x y H j x i i j (4) In genera, the q x that maximizes the error exponent is given by a distribution concentrated on a thin spherica she ]. However, choosing q x as the Gaussian distribution ends itsef usefu for simpified expressions and a convenient ower bound on the E o (and consequenty an upper bound on the probabiity of error). Therefore, we consider the distribution of the x to be zero mean Gaussian with covariance Q E xx ]. Since, we assume that channe knowedge is not avaiabe at the transmitter, Q SNR I. The function E o is reated to the queuing parameters as given beow. By making use of the formaism in 3], each user who communicates with the receiver is modeed as a job in a processor sharing queue. For a queuing deay anaysis, we need to express random message arriva parameters and channe parameters in terms of demand and suppy. By taking ogarithm on both sides of () and rearranging the expression, we can write the demand per unit bandwidth for a toerabe error probabiity and message ength (og K) as, S og P e + ρ og K (5) The remaining term after taking ogarithm on both sides of (), ( i E o) can be treated as accumuated service. The service rate at time t can be obtained by evauating the error exponent at time t and scaing it with the bandwidth. Thus, the units of the service rate wi be bits/sec. The service rate at time t with joint decoding can therefore be written as, φ(u) WE o (ρ, Q, H u ) (6)

3 In 3], since SUD was considered, the sum service rate was the sum of the service requirement of the individua users. Assuming identica channe distribution for a users, the sum service rate for SUD was given as φ(u).w.e o (ρ, Q, H ). However, since we are considering joint decoding, a the users are decoded together and hence we compute the joint decoding error exponent E o (ρ, Q, H u ). The average deay in servicing these users(or the average transmission duration) is obtained by Litte s Law ] which reates the deay to the average number of users in the system and the arriva rate of the users. The average deay is given by, D EU]/, (7) where the distribution of the number of users is given by 3], with, φ! (u) Pr{u jobs in the system} ν φ(ν) and C + Cφ! (u) (ES])u (8) (ES]) u /φ! (u) (9) u A condition for stabiity of the system in this case can be derived in a manner simiar to 3] and is not shown here. The oading of the queue is defined as, ES]. The expectation in (9) is over the distribution of the number of codewords for different users. If we make a simpifying assumption that the cardinaity of the set of codewords for each user is K and substitute for ES] from (5), ρ og K og P e ] () The average deay can be numericay evauated by computing the error exponent averaged over various channe reaizations for a given probabiity of error, arriva rate and number of codewords. In order to obtain a cosed form expression for the deay, we need a cosed form expression for the error exponent. The foowing section makes use of the scaing aw in the imit of arge SNR in order to obtain an approximate expression for the average queuing deay. IV. ANALYTICAL APPROXIATION We now derive anaytica approximations for the average deay for a IO AC. A cosed form expression for the service rate and consequenty the error exponent term E o is the key to a cosed form expression for the deay. The error exponent term E o is akin to the sum rate capacity expression of a IO AC except for a scaing of the SNR term. By making use of Jensen s inequaity in (3), it can be easiy shown that the error exponent for the IO AC with joint decoding is given by ], E o E H ρ og I i SNR N + H iqh i +ρ, () where the free parameter ρ can be numericay seected over, ] to either minimize deay 3] or to minimize the probabiity of error 9]. Numerica anaysis indicates that ρ maximizes the error exponent and hence we set ρ.we now make use of scaing aws in the imit of arge SNR to obtain cosed form expressions for E o. The scaing aw for a fast fading IO AC was mentioned in ] as, im E u H og SNR I + SNR H ih i i SNR min (u, N) og () The arge SNR assumption is usefu in obtaining performance imits for characterizing the queuing deay. Let us define by W.N. og SNR, α t W.. og SNR (3) In terms of α t and, the service rate can be expressed as, φ(u) WE o (ρ, Q, H u ) SNR W min (u, N) og ( + ρ) min (u.α t, ) (4) In the imit of arge SNR, we negect the scaing of the SNR term by ( + ρ) in (4). Theorem : The average deay for a IO AC under fast fading with no CSIT and fu CSIR is approximatey given by D ] for N ( ) for <N α t (5) The approximation is accurate for arge SNR and if the stabiity conditions are satisfied, viz., Proof: The average deay is given by <min (α t, ) (6) D u u (ES]) u Cφ! (u) (7) By making use of (4), the deay expression can be simpified further. Since the scaing aw for the service rate depends on the number of users, we consider two cases, N<and δ < N (δ +)N, δ Z +. Case A: If N<, and φ (u) u φ! (u) α u r (8) ( ) j C + (9) j

4 The convergence of the infinite geometric series in (9) is vaid when / <. The average deay is therefore, D ( ) u u. j ( ) ] Case B: δ N,(δ +)>N, δ Z + If u>δ, φ! (u) W og SNR ] δ ( δ! WN og SNR WN og SNR ] u δ! δ N δ ) u δ () α u r δ!β δ () where we have denoted (/N) as β. For u<δ, φ! (u) W og SNR ] u.u! αt u u! () The normaizing constant C can therefore be written as, δ ( ) j C + j! + ( ) j δ!β δ (3) j α t ( ) exp + α t δ!β δ ( jδ+ ) δ+ ( ) (4) The approximation of the first power series summation in (3) by an exponentia series is vaid for /α t <. The convergence of the second term in power series expansion in (3) is vaid for / <. The average deay can therefore be written as, D δ C. u (/α t ) u + u! u uδ+ ( ) C. exp + α t α t ( ) δ+ ( ) ] δ+ (δ +) δ ] u (/ ) u δ! β δ δ!β δ ( ) α t (5) By combining the criteria for stabiity in Case A and B, we get the stabiity condition as <min (α t, )). Comment : If < N, the deay is inversey proportiona to the number of transmit antennas. Thus, increasing the ] number of transmit antennas decreases the deay. This decrease in deay is unike the non-monotonic behavior seen if singeuser decoding is used 6]. Singe-user decoding imits the maximum service rate to 3]. Increasing the number of transmit antennas with SUD decreases the service rate further. In contrast, with joint decoding, the deay decreases with addition of antennas at user terminas as seen in (5) and (). Therefore, to harness the benefit of mutipe antennas for deay in a mutipe access setting, joint decoding is critica. The pot of the average deay v/s the number of transmit and receive antennas is shown in Fig. and Fig.. Deay N Fig.. Pot of deay v/s number of receive antennas, N with 8 transmit antennas at SNR of 4dB and a desired probabiity of error of 5. The soid ine is numerica evauation whie the dashed ine is anaytica approximation. Deay Fig.. Pot of deay v/s number of transmit antennas, with 8 receive antennas at SNR of 4dB and a desired probabiity of error of 5. The soid ine is numerica evauation whie the dashed ine is anaytica approximation.

5 V. NUERICAL EVALUATION In Figs. and, the pots of the average deay v/s the number of transmit and receive antennas indicate that the approximation is vaid for high SNR. In Fig.3,we pot the deay v/s SNR in db for a and a 4 4 system with a desired probabiity of error of 5. The arriva rate chosen was. which satisfied the stabiity condition. As expected, we observe that the deay decreases with increase in the SNR due to the inverse reationship with number of antennas (). The cosed form expression is accurate for medium to arge SNR. IfN>, the same trend is seen. Deay.5.5 SNR 4dB SNR db 6 4 P e Deay 8 6 Fig. 4. Pot of Average Deay v/s Probabiity of error for a 4 4 system at db and 4dB SNR. The soid ines are numerica evauation whie the dashed ines are the anaytica approximation SNR in db Fig. 3. Pot of Average deay v/s SNR in db for a and 4 4 system. The soid ines are numerica evauation whie the dashed ines are the anaytica approximation.. In Fig. 4, we pot the average deay v/s the probabiity of error by making use of (). Increasing the specification on the probabiity of error causes the deay to increase. An arriva rate of. was chosen. With decrease in P e, the oading of the system, increases resuting in an increase in the vaue of the numerator in the approximate expression (), (5). The probabiity of error term appears in the denominator too. But, it is scaed by the arriva rate and is negigibe compared to the term that is asymptotic in SNR. Intuitivey speaking, a ower vaue of probabiity of error means that the receiver shoud accumuate successivey more vaues of E o to meet the specification. This woud mean that the transmitter has to send more bits and consequenty the increase in the deay. Foowing the trend of Fig., increasing the SNR, decreases the deay and the approximation becomes more tighter. If N>, the trend is again simiar, but the approximate expression is tighter as increases. VI. CONCLUSION In this paper, we derived a simpe cosed form approximation for the queuing deay in a AC with mutipe transmit and receive antennas. Our resuts show that the average deay is inversey proportiona to the minimum of the number of transmit and receive antennas. Comparison with numerica anaysis indicates that the proposed anaytica approximation of the deay is accurate for medium and arge SNRs. The existence and design of coding strategies that achieve the performance predicted by using mutiuser decoding in the proposed scenario shoud be investigated in future work. Future work shoud aso consider other arriva processes with ong range dependence. ACKNOWLEDGENT This work has been supported in part by NSF under grant CCF REFERENCES ] S. Shenker, C. Partridge, and R. Guerin, Specification of guaranteed quaity of service, in RFC : Internet Eng. Task Force, Sept ] J. Kurose and K. Ross, Computer Networking: A Top Down Approach Featuring the Internet. Addison-Wesey,. 3] I. Teatar and R. Gaager, Combining queueing theory with information theory for mutiaccess, IEEE Journa on Seected Areas in Comm., vo. 3, pp , Aug ] R. Prakash and V. V. Veeravai, Centraized wireess data networks with user arrivas and departures, IEEE Trans. on Info. Theory, vo. 53, pp , Feb. 7. 5] P. Eia, S. Kittipiyaku, and T. Javidi, On the responsiveness-diversitymutipexing tradeoff, in WiOpt, Apri 7. 6] S. N. Kizhakkemadam and D. Rajan, Queuing aspects of mutiantenna mutipe access channes, in IEEE Gobecom, (San Francisco), Nov.- Dec. 6. 7] T.. Cover and J. A. Thomas, Eements of Information Theory. Wiey, 99. 8] F. P. Key, Reversibiity and Stochastic Networks. New York: Wiey, ] R. Gaager, Information Theory and Reiabe Communication. New York: Wiey, 968. ] I. E. Teatar, Capacity of muti-antenna Gaussian channe, European Tran. on Teecom., vo., pp , Nov./Dec ] D. Bertsekas and R. G. Gaager, Data Networks. Engewood Ciffs, NJ: Prentice Ha, ed., 99. ] T. Guess and. Varanasi, Error exponents for the Gaussian mutipeaccess channe, in IEEE ISIT, p. 4, Aug. 998.