Delay Minimization for Random Scheduling In Centralized Wireless Networks

Xin Liu Broadcom

14 Cliffwood Ave, Matawan, NJ 07747

Email: xinliu@broadcom.com

Athina P. Petropulu ECE Department

University of Rutgers Piscataway, New Jersey 08854

Email: athinap@rutgers.edu

Saswati Sarkar ECE Department

University of Pennsylvania Philadelphia, Pennsylvania 19104

Email: swati@ee.upenn.edu

Abstract-We consider a centralized network scheme, where all nodes communicate with a base station (BS). The communication happens in a time slotted fashion and the BS has multi-packet

reception capability. Assuming that the success probability of transmissions is always 1 (perfect reception), we prove a lower bound on the delay performance for arbitrary policy. We then study the imperfect reception case in which the success prob­ability of transmissions is not always 1. A convex optimization problem is proposed, which can minimize the upper bound on the expected delay of random scheduling (RS). An approximation and a lower bound on the delay of RS are also developed for the case in which the BS can support simultaneous transmission of up to two users. We implement a recently proposed algorithm that separates multiple simultaneous transmissions in a blind fashion, and show that the approximated delays and the delay bounds match the simulation results very well.

Index Terms- Wireless Networks, Random Scheduling, Delay Analysis, Multi-packet Reception

I. INTRODUCTION

1 We consider a centralized network scheme, in which all nodes communicate with a BS and the communication occurs in a time slotted fashion. The BS can resolve simultaneous transmissions of multiple users. We consider the general multi­packet reception case, i.e. the success probability of a node varies depending on which nodes transmit along with that node. The same scenario operating under the ALOHA pro­tocol, i.e., each node transmitting with a fixed probability and the BS having multi-packet reception capability, was employed in [2], where the stability region and delay were analyzed. In this paper we consider the problem of scheduling of nodes that will transmit simultaneously. In particular, we consider an RS scheme. In RS, once the scheduling probability is calculated by the BS, for each time slot the scheduling decision is made independently at each node. Thus, unlike the maximum weight scheduling (MWS) [5], [4], there is no need to exchange scheduling information between the BS and the mobile nodes in each slot, nor is global knowledge of the queueing length in the entire network required.

The main contributions of this chapter are the following (i) Assuming the success probability of transmissions is always 1, we provide a lower bound on the delay performance for

'This work has been supported by the National Science Foundation under Grant CNS-09J6947.

an arbitrary scheduling policy. (ii) For the cases in which the success probability of transmissions is not always 1, we propose a convex optimization formulation that can minimize an upper bound of the expected delay by determining the optimal scheduling probability of each node. Also the op­timization formulation guarantees that RS can stabilize any packet arrival rate within the stability region. (iii) We develop an approximate expression and a lower bound for the expected delay of RS for the case in which two users can transmit simultaneously.

Related work The MWS approach [5] has received a lot of attention. In any given time slot, MWS schedules a set of links with the longest accumulated queuing length. The scheduling should satisfy the interference and conflict requirements and the weighted sum of the queueing length of the chosen set should be the largest among all such sets. MWS is throughput optimal and can achieve the stability region of a system. The delay property of MWS is studied in [3], [4]. To combat the high computational complexity of MWS, approximations were proposed in [6]. MWS and its approximations require knowledge of queueing length of the entire network, which is hard to obtain in a real implementation.

The throughput analysis of RS with multi-packet reception capability can be found in [7], [8]. It is shown in [7], [8] that RS can achieve the stability region of a generalized wireless network with multi-packet reception capability. However, the delay property was not considered in [7], [8]. Moreover, in the application section of [7], [8], the authors analyzed the capacity of the Manhattan network and ring topology network by assuming that either the success probability of a node depends only on the cardinality of the transmission set, or the success probability of a node, when it transmit along with different nodes, is either 1 or O. Throughput maximization with delay constraints was proposed in [10]; that work did not consider any multi-packet reception capability. The stability region and delay of ALOHA with multi-packet reception can be found in [2]. However, an ALOHA network cannot achieve the stability region of the centralized network when the total number of users is more than 2. The delay analysis in [2] was restricted to the two user case.

978-1-4673-3140-1/12/$31.00 ©2012 IEEE

II. SYSTEM MODEL

Suppose there are N users in a centralized wireless network, and let Sm, m = 1, ... ,M denote independent sets of users who can transmit simultaneously with high success probability. In each time slot, the nodes with non-empty transmission queue within Sm will be selected to transmit, while other nodes remain silent. This can be achieved in a non-centralized fashion as follows [8], [9]. All nodes have the same random number generator "seed". At the beginning of a time slot, each node generates a random number, i.e., u. Note that this number will be the same for all nodes since all nodes are using the same seed. Assume that the probability of selecting Sm is Pm (the way to determine Pm will be described later). If u < PI, the nodes in Sl will transmit; if L;:�l PI :s; u < L;:lPI, m > 1, the nodes in Sm will transmit. After that all the nodes update their seed simultaneously.

We make following assumptions: (AI) The packet will arrive at the beginning of time slot and leave and the end of time slot. There is at most one arrival or departure per time slot for each node. (A2) The queue size at slot t is measured after the arrival and before the departure. (A3) Packets are eligible for transmission in the same time slot in which they arrive. (A4) Both the arrivals and the departures are Bernoulli distributed. (A5) The success probability of a node varies depending on which other nodes transmit at the same time. (A6) For the simultaneous transmission of multiple users, the success of a user's transmission is independent of the transmission outcome of other users.

III. DELAY MINIMIZATION WITH PERFECT RECEPTION

In this section we will determine the scheduling probability of Sm to minimize the expected packet delay. We start with the simplest possible case in which the BS can support simul­taneous transmission of two users and in which the packets can be correctly received with probability 1 (perfect packet reception). The imperfect reception case will be discussed in Section IV.

Let r i denote the packet arrival rate of node i, and qi the probability of departure (i.e. successful transmission) of node i when node i is active. The network can support simultaneous transmission of n � 2 users, where n is the maximum number of simultaneous transmissions allowed. The success probability for the n users is 1. In this case the number of independent sets is M = (�). Let P = [PI, . . . ,P M ] T , where

Pm denotes the probability of choosing the set Sm, and Cm the indicator vector of Sm, i.e. if i E Sm, the i-th component of Cm is 1; otherwise it is O. There are exactly n components equal to 1 in each Cm. The delay optimization problem can be expressed as

(1)

M

s.c. qi > ri; q = L Pmcrn; IT P = 1; O:S; Pm :s; 1. (2) m=l

The stability region discussion of a generalized wireless net­work with multi-packet reception can be found in [8]. From the discussion of [8] we can see that the feasible region of the this convex optimization method given by (2) guarantees that RS can stabilize any packet arrival rate within the stability region.

A. Lower bound for any policy

Proposition 1: In a centralized network of N users, in which the BS can support simultaneous transmissions of n users, the expected delay of an arbitrary policy is lower bounded as:

",N 1ro-r2 D> L..-,= 0 , • - N ( N ) 2 Li=l ri n - Li=l ri

(3)

Proof: See Appendix A. We should note that this lower bound for an arbitrary

policy is different from that proposed in [3], [4]. Here, we consider a centralized network and any n users can transmit simultaneously. Unlike [4], for each link l we do not have a exclusive set XI in which no more than one link can be scheduled at any given slot. The above lower bound depends on the packet arrival rate ri, whereas the bound of [3] does not.

IV. DELAY MINIMIZATION WITH IMPERFECT RECEPTION

Next, we consider a more realistic case, in which the success probability of the simultaneously transmissions is not always 1. We assume that the BS can support simultaneous transmissions of n = 2 users. Let Pi,j be the probability of scheduling nodes i and j, and Pi,i the probability of scheduling node i only. Let Bi,j be the (i , j)-th element of B, denoting the success probability of node i when node i and j transmit simultaneously and Bi,i be the success probability of node i, when only node i transmits. We assume Bi,i > Bi,j, since without interference from node j the success probability of node i should be higher. The probability of success of node i can be approximated as:

N qi � L Bi,j Pi,j (4)

j=l which is an underestimate of the actual success probability. To see why, suppose that in some slot, nodes i and j were selected to transmit, but the transmission queue of node j is empty. In that case, node j would remain silent and only node i would transmit. Then, the success probability of node i would Bi,i instead of Bi,j. Hence (4) is an underestimate of the success probability of node i.

The delay minimization problem can be formulated as

(5)

N N L L Pi,j = 1; 0 � Pi,j � 1; Pi,j = Pj,i i=l j=i

By solving this problem we can get the optimal scheduling probability of each independent set, which can minimize the delay of packets in the network. It is worth noting that although the BS can support simultaneous transmission of two users, it is not necessary that we have to schedule two users for every slot. If the success probability of simultaneous transmissions of two users is low, we should schedule only one user in each slot.

A. Approximation for the delay

In this section we introduce an approximation for the packet delay of RS. Eq. (4) gives a lower bound of the success probability of node i by assuming that the nodes are always active. Let ki be the active probability of node i. The success probability of node i can be approximated as

qi � ( . t . Bi,jPi,jkj + Bi,iPi,j(1- kj)) + Bi,iPi,i. (6) J=l,J,ot

At the same time, if the queues are stable, we have ri = kiqi' Let

9 =k (( � B ·P k + B B '(l -k )) + B 'P' ) to to � 1,,) 1.,,) J 1,,1, 1,,) J 1,,1, 1"Z j=l,jf-i

- ri , (7)

By solving gi = 0 for i = 1, . . . , N we can calculate the active probability ki. The schedule probability Pi,j and

Pi,i are obtained by solving the optimization problem (5). After calculating the active probability ki, we can obtain

qi based on qi = rj ki. The packet delay equals D = L:� riDj (L:� ri) ' where Di = (1 - ri)/(qi - ri)' For N = 2, it can be proven that there is only one solution

for ki within the region [0, 1] (see proof in Appendix B). For N > 2, we use simulation (in section V) to show that the approximated delay is very close to the expected delay.

B. Delay lower bound of RS for imperfect reception

In this section we derive a lower bound for the expected delay of RS by assuming that up to two users can transmit simultaneously.

Let bm = [b1 Im, . . . , bN1mV, where bi1m is the success probability of node i if i E Sm, when the nodes within Sm transmit simultaneously. If i 1:- Sm, bilm = O. Let Pm be the probability that we schedule the independent set Sm; and define P{.} = Pmlsm=O and bO = bmlsm=o. According to this definition P{ i, �} is the probability that we schedule nodes i and j; Ilb{i} III is the success probability of node i when only node i transmits and Ilb{i,j}III is the sum of the success probability of node i and j, when these two nodes transmit simultaneously. We further assume that the success probabilities for the simultaneous transmission of two nodes are high, i.e.,

• A7) Ilb{i} III < Ilb{i,j} III for i,j = 1, . . . , N and i � j.

The assumption (A 7) indicates that the success probability of node i when only node i transmits is smaller than the sum of the success probability of node i and j when these two nodes transmit simultaneously.

Before we start to derive the lower bound, we introduced three basic constraints that have to be satisfied by a stable queue.

(8)

ri+ rj �P{i,j} Ilb{i,j}III + ( L Pm -P{i,j}) Ilb{i}III tESm

+ ( L Pm -P{i,j}) Ilb{j} III for i � j . (9)

JESm N M

Lri � L Pm IIbml11 i=I m=l

(10)

(8) is the requirement for the packet arrival rate of node i. The success probability of node i when only node i transmits is Ilb{i} Ill' The success probability of node i will decrease if node i and j transmit simultaneously. Therefore if queue i is stable, (8) has to be satisfied.

(9) is the sum rate requirement for node i and j. Due to assumption (A7), if we schedule node i and j, the sum of the departure rates of nodes i and j cannot be greater than

lib {i,j} Ill; if either node i or j does not transmit, the sum of the departure rates will only decrease. If we schedule node i and l, l � j, the "best" case for node i is for I to keep silent and only node i to transmit. In this case, the success probability of node i is Ilb{i}III'

(10) is the sum rate requirement. Because of assumption (A 7), if we schedule nodes within Sm, the largest possible expected sum of delivered rate is IlbmllI. Given that Pm is the probability of scheduling Sm, the total throughput of this system cannot exceed L::;;= 1 Pm II bm Ill' In order to maintain the stability of the system, (8), (9) and (10) are necessary but not sufficient.

With these two restrictions we derived the lower bound of expected delay for RS, which is given in proposition 2.

Proposition 2: In a centralized network of totally N users, if we allow two users to transmit simultaneously, the expected delay of RS is no smaller than maxur, fn. R is the the optimal value of the problem

(11)

s.c. qi ?: ri, qi = L Pm Ilb{i} Ill' (12) iESm

li+ lj ::;P{i,j} Ilb{i,j}lll + (L Pm -P{i,j}) Ilb{i}lll tESm

+ ( L Pm -P{i,j}) Ilb{j} III for i -1= j , (13) JESm

N M

L 'i::; L Pm Ilbmlll i=l m=l

(14)

1 T P = 1; 0::; Pm ::; 1 m = 1,2, . . . ,M. (15)

",N 2

f* - L..,i-l 'i - 'i (16) 2 - 2L!1 Ii (U* - L!l Ii) where U* is the optimal value of the problem

M maxU = L Pm Ilbmlll p,q m=l

(17)

s. C. qi � 'i, qi = L Pm Ilb{i} Ill' (18) iESm

Ii + Ij ::; P{i,j} Ilb{i,j} III + (L Pm -P{i,j}) Ilb{i} III tESm

+ ( L Pm -P{i,j}) Ilb{j} III for i -1= j , (19) ]ESm

N M

L Ii ::; L Pm Ilbmlll ; e p = 1; 0::; Pm ::; 1 (20) i=l m=l

Proof: See Appendix C.

C. Extension to the general multi-packet reception with im­

perfect reception

In this section we will expand our discussion to the cases in which the BS can resolve n > 2 simultaneous trans­missions, where n is the maximum number of simultane­ous transmissions that the BS can support. We use Sm to denote the independent set m, where m = 1, . . . , M and M = L�=l (�). M is the total number of independent sets.

bm = [bllm"' " b N lm ] T , is the success probability vector defined in Section IV-B. Moreover, if Sm' C Sm and Sm' -1= 0, for any j E Sm' it holds that bj1m, � bj1m. Sm' C Sm means there are fewer nodes in set Sm' than in Sm. Therefore, if the nodes in Sm' transmit simultaneously, for each node the mutual interference is smaller than that in Sm, which leads to the higher success probability of the nodes in Sm"

The delay minimization problem is formulated as

N

N1 """ Ii (1

. _-:.

i) min f (q) = �,.,--- � , p,q Li=l Ii i=l q, t (21)

M

s.c. qi > 'i, q = L Pmbm, ITp = 1, 0::; Pm::; 1. m=l

It is worth noting that q = L�=l Pmbm is an underestimate of the success probability of the nodes. As explained before, when Sm is scheduled, the nodes i E Sm might be inactive, therefore node i will not transmit. In this case the success probability of an active node j E Sm should be higher than bj1m. Hence (21) minimizes the upper bound of the packet delay.

From the stability region analysis of [8], we can see that the feasible set of this convex optimization problem guarantees that the proposed RS can stabilize any packet arrival rate in the stability region. Therefore the proposed RS achieves the stability region of the centralized system.

V. SIMULATION

In this section we demonstrate the validity of the approxi­mation of Section IV-A and the lower bound of RS in Section IV-B via computer simulations. We first measured the success probability of each node by letting two or three users transmit simultaneously. The user signals contained carrier frequency offsets (CFO's) (see [1]) that were randomly in the range r 0, O.��l ]. The user delays were uniformly distributed within to, Ts] . The input signals were differential quadrature phase shift keying (DQPSK) containing 1024 symbols. The IOTA pulse shape waveform with time support [-2Ts, 2Ts] was used for transmission. The sampling rate of the system was M = 32 samples/symbol; since the time support of the pulse was [-2Ts,2TsJ, this corresponds to M' = 4M + 1 = 129 samples per pulse. The blind source separation algorithm of [1] applied to recover the trnsnmitted packets. The estimation results are averaged over 100 independent channels, and 10 Monte-Carlo runs for each channel. The success probability of a node was computed as Ns/Nt, where Ns was the number of packets successfully received by BS, and Nt was the total number of packets transmitted.

Due to space limitations we consider the asymmetric case only. We set the total number of users N = 10 and SNR 16 dB. The channel amplitudes of the 10 users were [0.7494, 0.9934, 0.2188, 0.2516, 0.9123, 0.6804, 0.2276, 0.4906, 0.6324, 0.4009] . The success probability matrix B was computed as described above, and was subsequently used to calculate the optimal scheduling probability via the optimization method of (5). The expected delay performance of RS and the approximation are given in Fig. 1.

The match between the proposed approximation and the simulation is pretty close. Regarding the lower bound, when when the packet arrival rate is low and medium low, the lower bound is tight, while when the packet arrival rate is high, the lower bound becomes loose. In the same figure, we plot the delay performance of uniform scheduling for comparison. We observe that the delay of uniform scheduling is higher than that of the proposed method because the proposed scheduling

6°ir===�==�====�==�---'---'----fl�1 ---B--- Simulation optimal scheduling 50 --- Approximation

--- Lower bound --+- Upper bound

40 -E- Simulation uniform scheduling

20

10

OL-__ � __ ���L-__ � __ � __ �==�� __ � 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Packet arrival rate

Fig. 1. Total delay performance comparison for the asymmetric case, N = 10 and n = 2.

method tends to schedule the sets of nodes for which the success probabilities of simultaneous transmissions are high.

Next, we increase the SNR to 25 dB. The channel amplitude of each user is set the same as in the previous case. With higher SNR, the BS can successfully receive simultaneous transmis­sions of three users. The corresponding delay performance is plotted in Fig. 2. In the same figure we also plot the delay of uniform scheduling for comparison. We observe that by using the proposed optimization method the expected delay is much lower than that of the uniform scheduling.

-B-- Simulation optimal scheduling --+-- Upper bound ___ Simulation uniform scheduling

50

>-21 40 '" "0 (ij � 30

20

10

oL-�==�==�--�==�==��==�==�==�. 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

Packet arrival rate

Fig. 2. Total delay performance comparison for the asymmetric case, N = 10, n = 3.

V I. CONCLUSIONS

We have analyzed the delay properties of RS in cellular networks by assuming that the BS has MPR capability. For the perfect reception case, we have provided a fundamental lower bound for the delay performance of an arbitrary scheduling policy. For the imperfect reception case, we have proposed a convex optimization problem that can minimize the upper bound of the expected delay of RS by determining the schedul­ing probability of each independent set of nodes. We have also developed an accurate estimate and a lower bound for the delay of RS for the case in which the BS can successfully receive simultaneous transmission of two users. It is worth

noting that RS does not require the BS to exchange scheduling information with the mobile nodes, nor does it need global knowledge of the queueing length in the entire network. These properties make RS an ideal scheduling policy for cellular networks.

ApPENDIX A

PROOF OF THE PROPOSITION 1

Proof: Due to assumptions AI) and A2), it holds that

Qi(t + 1) = Qi(t) - Di(t) + Ai(t + 1) where Q, A, D denote instantaneous queue length, arrival rate and departure

rate, respectively. Let us define V (Q(t)) = (2::1 Qi(t)f. Takin expectation and setting to zero we get

E { (t, Di(t) -t, Ai(t + 1)) t, Qi(t)}

� �E { (t, A,(t+ \) -t,D,(t)) , } . (22)

Let E {Ai(t)} = 'i. When the system is stable, the packet arrival rate should be equal to the packet deliver rate, i.e.,

E {Di(t)} = E {Ai(t)} = 'i. Because of AI), Ai(t) can be either 0 or 1; hence Ai(t)2 = Ai(t). Due to A4), it holds that E {Ai(t)Aj(t)} = lilj for i i- j. Based on the above, using Jensen's inequality, and noting Ai(t+ 1) is independent of Dj(t) and Qi(t), we get

Since the packet arrival rate is independent of queueing length, we can get

E {t, Ai(t + 1) t, Qi(t) } = t, liE {t, Qi(t)} (24)

Since in each time slot the BS can at most receive n packets, it is obvious that 2::1 Di(t) � n.

E {t, Di(t) t, Qi(t)} � nE {t, Qi(t) } (25)

By substituting (24) and (25) into (23), we have

E {tQi(t)} 2': 2::1 Ii;; I; . (26) i=l 2 ( n -2:i=l Ii)

By applying Little's law, the expected delay lower bound of an arbitrary policy is that of proposition 1.

ApPENDIX B

For N = 2 by forcing gi = 0 i = 1,2, we have

kl(Bl,2Pl,2k2+Bl,lPl ,2(1-k2)+Bl,lPl ,1)-rl = 0 , (27a)

k2(B2,lPl ,2kl +B2,2Pl,2(1-kI)+B2,2P2,2)-r2 = O . (27b)

From (27a) we get

k _ rl 1- Bl,2Pl,2k2 + Bl,lPl ,2(1 -k2) + Bl,lPl ,l Substituting (28) into (27b) we obtain

(Bl,2 - Bl ,1)(Pl ,2 + P2,2)B2,2Pl ,2k� + ((Bl ,lPl ,2 + Bl,lPl ,l - rI)B2,2Pl,2 + (Pl ,2 + Pl,1)Bl,lB2,2P2,2 - (Bl,2 - Bl,I)Pl ,2r2

(28)

+ B2,lPl ,2rdk2 - Bl,1(Pl ,2 + Pl,l )r2 = 0 . (29)

By forcing k2 = 0, the left hand side of (29) is -Bl,1(Pl ,2 + Pl,dr2 < 0; by forcing k2 = 1, the left hand side of (29) is equal to

Sm transmit simultaneously, otherwise E {L!l Di(t)} <

Ilbm IiI- Given the probability of scheduling Sm to be Pm, E {L!l Di(t)IQ(t) = Q} :s; L�=lPm Ilbmlll· Moreover,

(18), (19) and (20) are the necessary conditions for the scheduling probability Pm to maintain the stability of the system. Therefore,

where U* is the optimal value of the problem (17). Substitut­ing (32) into (31) we have

E {t, Di(t) t, Qi(t) } :s; U* E {t, Qi(t)} . (33)

By substituting (24) and (33) into (23), we can prove D ?: 12'

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[2] Y. Naware, G. Mergen, and L. Tong, "Stability and delay of finite­user slotted ALOHA with multipacket reception," IEEE Transactions The solution of (5) satisfies that Bl,2Pl ,2 + Bl,lPl ,l > rl

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ApPENDIX C PROOF OF THE PROPOSITION 2

Proof: Let D be the expected delay of RS. We will prove that D ?: Ii and D ?: 12'

D ?: Ii: Ilb{i} III is the success probability of node i when only

node i transmits. And the success probability of node i will decrease if node i and j transmit simultaneously. Therefore,

LiEs=Pm Ilb{i}lll is the upper bound of the success proba­bility of node i. Moreover, (12), (13) and (14) are the necessary conditions for the scheduling probability Pm to maintain the stability of the system. Hence, Ii is the lower bound of the expected delay of RS.

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for all(t t-l t-l

= L E {tDi(t)IQ(t)=Q}tQiP {Q(t)=Q} for all(t t-l t-l

(31)

Because of assumption (A7), when we schedule Sm in

slot t, E {L!l Di(t)} = Ilbmlll if all the nodes within