Index TermsâUnderwater acoustic communication (UWAC), broadcast scheduling problem (BSP), spatial-reuse scheduling, topology-transparent scheduling.
Robust Spatial Reuse Scheduling in Underwater Acoustic Communication Networks Roee Diamant, Ghasem Naddafzadeh Shirazi and Lutz Lampe University of British Columbia, Vancouver, BC, Canada, Email: {roeed,naddaf,lampe}@ece.ubc.ca Abstract—In this paper we address the problem of spatialreuse scheduling for underwater acoustic communication (UWAC) networks that support high traffic broadcast communication and require robustness to inaccurate topology information. To this end, we derive a broadcast scheduling algorithm that combines topology-transparent and topology-dependent scheduling methodologies to achieve high-throughput in static and dynamic topology scenarios. While we focus on scheduling in UWAC networks, our approach can also be adopted for broadcast scheduling for radio ad-hoc network if robustness to topology uncertainties is desired. Simulation results for typical UWAC scenarios demonstrate that our protocol achieves high throughput and provides robustness to outdated topology information in dynamic topologies. Index Terms—Underwater acoustic communication (UWAC), broadcast scheduling problem (BSP), spatial-reuse scheduling, topology-transparent scheduling.
I. I NTRODUCTION Scheduling in underwater acoustic communication (UWAC) networks is challenging mostly due to the use of acoustic signaling. The low sound speed in water (approximately 1500 m/sec) leads to long propagation delays and necessitates large guard periods, and considerable bandwidth limitations exist due to the frequency and transmission range tradeoffs [1]. Scheduling is also limited by half duplex communication, due to design constraints of acoustic transducers [2]. Moreover, the time-varying characteristic of the underwater acoustic channel [3] and frequent mobility in the network result in a timevarying topology, which requires scheduling schemes to be robust to topology uncertainties. In this paper we consider the problem of resource assignment through scheduling in high-traffic UWAC networks which support broadcast communications. This is required for applications such as sharing of navigation information, system control, sending distress signals, etc. Most available scheduling algorithms for UWAC networks rely on multiple access with collision avoidance (MACA) protocols [4], where request-to-send (RTS) and clear-to-send (CTS) packets are used. However, since nodes detecting an RTS or CTS packet should be kept silent, channel utilization decreases. As shown in [5], the number of silenced nodes grows quadratically with the communication range and the MACA protocol becomes more and more conservative. The low channel utilization of MACA is compounded in UWAC networks, where the long propagation delay necessitates longer silence periods [4]. A different approach is to apply spatial-reuse over the time division multiple access (TDMA) protocol, where significant
improvement in channel utilization is possible. Spatial-reuse is an appealing technique in UWAC networks where range and frequency dependent absorption loss [3], and acoustic non-line-of-sight scenarios lead to sparse network graphs. For this reason, we recently presented a heuristic spatial-reuse TDMA schedule for broadcasting in UWAC networks based on the hop-distance between network nodes [6]. Optimization of spatial-reuse TDMA for broadcast communication is also known as the broadcast scheduling problem (BSP). Typically, in the BSP, first the TDMA frame length is minimized and then channel utilization is maximized for the given the frame length using spatial reuse techniques [7]. Spatial-reuse TDMA scheduling schemes usually rely on accurate topology information being available to network nodes. However, this assumption may be violated when topology varies. As a result, several nodes might temporary hold conflicting topology information leading to packet collisions. Uncertainty of topology information can be regarded as a problem of robustness since we require a certain minimum performance to be achieved even under topology mismatch. The problem of robustness is considered in topology-transparent scheduling which does not depend on the instantaneous network topology. Topology-transparent spatial-reuse scheduling was pioneered in [8], which suggested a schedule based on the maximal degree of the network graph and an upper bound on the number of conflicts between any two nodes regardless of network topology. The work in [9] generalized this protocol and reduced the number of conflicts. In this paper, we propose a spatial-reuse TDMA scheduling protocol for UWAC networks, referred to as robust BSP (RBSP), that attempts to reconcile the seemingly conflicting requirements of high channel-resource utilization and robustness to network-topology information uncertainty. To achieve this goal, we combine the concepts of topology-transparent and topology-dependent scheduling. The former element ensures that topology information mismatch does not cause an uncontrolled amount of packet collisions. The latter component allows us to make use of additional spatial reuse in case of reliable topology information. Finally, we note that while we focus on the BSP for UWAC networks, our approach can also be adopted for the BSP for radio ad-hoc network if robustness to topology uncertainties is desired. II. S YSTEM M ODEL We consider UWAC networks with a fixed small to moderate number of nodes N , say N < 50, distributed over an area
of a few square kilometers. We require only a coarse timesynchronization between the nodes to establish a network-wide TDMA frame structure; that is, node clock offset and skew should be negligible compared to the propagation delay, which is on the order of 1 to 3 seconds for distances of 1 to 4 km. Due to the half-duplex characteristics of acoustic transducers [2], we do not allow nodes which are connected through onehop links to transmit simultaneously, i.e., we respect primary conflicts [10]. Due to the high channel attenuation in UWAC, secondary conflicts, i.e., when nodes located more than one hop away transmit simultaneously, are only important for nodes connected by two-hop links. We mitigate secondary conflicts making use of direct sequence spread spectrum (DSSS) signaling, which is part of the physical layer in many UWAC systems for improving the receiver side signal-to-noise ratio (SNR) and robustness to impulsive and narrowband noise [11]. We would like to mention, however, that our approach could also be applied to non-DSSS systems, by taking secondary conflicts into account similar to [10]. The output of our R-BSP protocol is an N × L spatialreuse TDMA (STDMA) binary scheduling matrix M such that node i, 1 ≤ i ≤ N , is allowed to transmit a single packet in time slot j, 1 ≤ j ≤ L, if and only if M i,j = 1, and L is the number of time slots in each TDMA frame. We introduce resource-allocation constraints that guarantee node i a minimal number of di time slots per time frame for transmission. That is, defining xi as the number of time slots assigned to node i within a time frame, we have xi ≥ di ,
i = 1, . . . , N .
(1)
Assuming that multihop routes have been established by a routing protocol, constraint (1) can be used to respect trafficflow demands and thus avoid bottleneck nodes. Defining routing variables θi,j , i, j = 1, . . . , N , with θi,j = 1 if node i is a relay for node j’s broadcast packets and θi,j = 0 otherwise (note that θi,i = 1), and di =
N X
gj θi,j ,
(2)
j=1
node i is guaranteed to transmit gi original packets per time frame through constraint (1). For measuring throughput, we define yi,j (T ) as the number of broadcast packets originated by node i and received by node j in T time slots. Assuming non-empty transmit queues at nodes, the per-node throughput is given by ρthrough =
N X N X 1 yi,j (T ) , T (N − 1)N i=1 j=1
(3)
j6=i
where we assume that the observation window T is much larger than the TDMA frame length L. For measuring scheduling delay, we consider the waiting period between two successive transmissions and the (multi-hop) propagation time to the destination. To formalize this, we define the binary indicator pi (t) which is 1 if and only if node i transmits
an original packet in time slot t. For time slot t such that pi (t) = 1, denote si,t as the time slot in which node i transmitted a previous original packet, i.e., pi (si,t ) = 1, where for time slot t′ in which i transmitted its first packet, si,t′ = 0. Furthermore, let Tpd(i, t) be the number of time slots it takes to successfully propagate the packet, averaged over all destinations. The scheduling delay of the protocol is then expressed by
ρdelay
T P pi (t) ((t − si,t ) + Tpd (i, t)) N X 1 t=1 = T P N i=1 pi (t)
(4)
t=1
Since both ρthrough and ρdelay depend on the network topology, it is hard to optimize them directly for broadcast communications. However, if flow demands are satisfied, channel utilization and thus availability of time slots to nodes is proportional to throughput. In addition, for a given L, channel utilization is inversely proportional to scheduling delay. Therefore, following, e.g., [7], we consider maximization of channel utilization N
ρavail =
1 X xi N L i=1
(5)
in the following formalization of the R-BSP schedule. III. T HE R-BSP S CHEDULE While spatial-reuse scheduling is promising for increasing channel utilization, it is sensitive to erroneous topology information. For example, consider two connected nodes that do not share the same frame length L due to flawed topology information. The result would be an almost random schedule and thus catastrophic packet collisions. Hence, a topologyindependent frame length is necessary to achieve robustness to uncertain topology information. We therefore suggest a new approach that combines an underlying skeleton schedule with a fixed frame length obtained from a topology-transparent schedule with the use of topology information alike in a topology-dependent schedule. A. Combining Topology-Transparent Dependent Scheduling
with
Topology-
We represent the set of possible node assignments to a time slot by the columns of matrix I ∈ {0, 1}N ×K such that for column j, In,j = Im,j = 1 only if nodes n and m can transmit in the same time slot respecting primary conflicts, where K is the number of possible such node assignments. The R-BSP is then solved by associating each column j of I with an integer aj ≥ 0, representing how many times each combination of node assignments is used in the resulting schedule M . Mathematically, defining vector a = [a1 , . . . , aK ]T , the vector x = [x1 , . . . , xN ]T of the number of time slots within a frame assigned to nodes is obtained by x = Ia .
(6)
Thus, (5) can be rewritten as (1 denotes the all-one column vector of appropriate length) 1 T 1 Ia . (7) NL To form matrix I such that the resulting R-BSP schedule will be robust to topology variations, we start from a topologytransparent schedule with an N × L skeleton matrix S whose element Si,j = 1 if and only if node i transmits in time slot j. First, the columns of S are rearranged to form a matrix S mod such that S mod r(j),j = 1 for a unique slot node r(j). Slot nodes serve as reference nodes for conflict removal and will always transmit in their time slots. Thus, the assignment of such nodes should be fair, and ideally network nodes will be uniformly selected as slot nodes. Algorithm 1 shows the pseudo-code of an L-step process in which slot nodes are selected in a round robin fashion to form S mod . In the lth step of Algorithm 1 slot node r(l) of the lth column of S mod is determined. We start with i = l mod N (line 5) and continue with i = (i + 1) mod N (line 15) until we find a column j in S which was not already assigned a slot node and for which Si,j = 1 (line 8). This column becomes the lth column of S mod (line 9) and we set r(l) = i (line 10). Note that this rearrangement of S into S mod does not affect channel utilization (7) of the topologytransparent schedule and is performed a-priori as it does not require topology information. Based on the rearranged skeleton matrix S mod , matrix I is constructed. This procedure, whose pseudo-code is shown in Algorithm 2, makes use of topology information (which, if erroneous, leads to an increased collision rate, but it does not cause a collapse of the schedule due to the underlying topology-transparent skeleton schedule). Considering the jth column of S mod , a list Tj consisting of all independent sets of the network graph which include r(j) is formed (line 2). To increase channel utilization while preserving as much as possible the structure of the topology-transparent schedule, the independent sets in Tj which include the largest number of pre-assigned nodes, Tjmod , are appended to I (lines 3, 4 and 6). Since there may be multiple independent sets chosen from Tj for each column j of S mod (i.e., |Tjmod | ≥ 1), the number of columns of I, K, is possibly larger than L. To maintain the deterministic frame size from the underlying topologytransparent schedule S mod , only one set of each Tjmod can be chosen to form the scheduling matrix M . This can be formulated by the constraint ρavail =
Aa = 1 ,
(8)
where A is an L × K 0/1-matrix such that An,m = 1 only if the mth column of I was derived from the nth column of S mod . The example shown in Figure 1 illustrates this process. Considered is a network of N = 6 nodes represented by the graph in Figure 1(a), and a given topology-transparent schedule S. First, S is rearranged into S mod using Algorithm 1 as shown in Figure 1(b), where the shaded entries represent slot nodes. Then, I is formed by expanding the columns of S mod utilizing
Algorithm 1 Rearranging matrix S into S mod 1: U := ∅ 2: for (l := 1 to L) do 3: {Determine the slot node} 4: FLAG := 0 5: i := l mod N 6: while FLAG = 0 do 7: for (j := 1 to L) do 8: if (Si,j = 1 and j ∈ / U) then mod := S {S , S j are the lth and jth 9: S mod j l l mod columns of S , S, respectively} 10: r(l) := i 11: U := {U, j} 12: FLAG := 1 13: end if 14: end for 15: i := (i + 1) mod N 16: end while 17: end for Algorithm 2 Determine I from matrix S mod 1: for (j := 1 to L) do 2: Tj : all independent sets in the network graph which include r(j) 3: Pj : all nodes i for which S mod i,j = 1 4: Tjmod : sets from Tj that include the largest number of nodes from Pj 5: end for 6: I := [T1mod , . . . , TLmod ]
the topology information according to Algorithm 2. In this example, there are two possible expansions for columns 5 and 8 of S mod . Finally, Figure 1(c) shows the masking matrix A for this example. B. Adjusting Flow Constraints in Robust Scheduling An R-BSP schedule with a fixed frame length L may not be able to satisfy the resource allocation constraints (1). Since the di in (1) are defined per time frame, one possible solution to avoid infeasibility and still maintaining the skeleton schedule is to repeat the schedule c times and use an extended frame length c · L, where c is a positive integer. By doing so, scheduling delay and throughput do not change. To formalize this, let vi be the pre-defined number of times that node i is selected as a slot node in one time frame of L time slots. Since a slot node always transmits in its designated time slot and xi ≥ vi , constraint (1) for node i is surely satisfied within a maximum of maxi (di )L di L ≤ vi mini (vi )
(9)
time slots. Using the upper bound from the right-hand side of i (di ) (9), we introduce a super-frame of c = ⌈ max mini (vi ) ⌉ sub-frames each consisting of L time slots in the R-BSP schedule. To
2
S=
6
1 4 3
5 (a)
A=
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
S mod
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 1
1 0 0 1 0 0
0 1 0 0 1 0
0 0 1 0 0 1
0 0 1 0 1 0
1 0 0 0 0 1
0 1 0 1 0 0
0 1 0 0 0 1
0 0 1 1 0 0
1 0 0 0 1 0
1 0 0 1 0 0
0 1 0 0 1 0
0 0 1 0 0 1
0 1 0 1 0 0
0 0 1 0 1 0
1 0 0 0 0 1
1 0 0 0 1 0
0 1 0 0 0 1
0 0 1 1 0 0
=
I=
1 0 0 1 0 0
0 1 0 0 1 0
0 1 1 1 0 0
0 1 1 1 0 0
z}|{ 0 1 1 0 0 0 0 0 1 1 0 1
8th column of S mod
5th column
1 0 0 0 1 1
1 0 0 0 1 1
0 1 0 0 1 0
z}|{ 0 1 1 1 0 0
0 1 1 1 0 0
(b)
(c)
Fig. 1: (a) Sample topology for a UWAC network. (b) Constructing matrix I using Algorithms 1 and 2 for the sample network. (c) Masking matrix used in (8).
ensure the flow constraints from (2), we can use
N P
gi for
0.9
C. Formalizing the Robust Scheduling Optimization Problem Using (8) and (9), the R-BSP can be formalized as T
max 1 Ia
(10a)
a
Aa = c · 1 ,
(10b)
K
a∈N . (10c) Since I = , the solution of (10) is aj(i) = c for j(i) being the index of the column of Timod with the largest number of non-zero elements, i = 1, . . . , L, and aj = 0 for the ˜ as the solution of (10) with elements remaining j. Denote a a ˜j being the number of times the jth column I j is used. The scheduling matrix M is then constructed as T1mod , . . . , TLmod
M = [I 1 . . . I 1 . . . I K . . . I K ] . | {z } | {z } a ˜1
(11)
a ˜K
For the example given in Figure 1 we choose columns 6 and 10 of I, since their one-norms are higher than for columns 5 and 9, respectively. Thus, the R-BSP schedule in (10) can be obtained without numerical optimization procedures. We note, however, that setting up I requires finding all independent sets of the network graph. The R-BSP in (10) is generic with regards to the matrix S that is used. However, channel utilization, scheduling delay and robustness are affected by the specific choice of the
Empirical Prob(ρdelay ≤ x)
maxi (di ) in (9).
1
i=1
0.8 0.7 0.6 0.5 0.4 0.3
TDMA T−BSP R−BSP
0.2 0.1 0 10
15
20
25
30
35
40
45
50
55
x
Fig. 2: CDF of ρdelay for fixed topologies, N = 8.
skeleton schedule. Due to space constraints, we next present simulation results for only one possible skeleton schedule, namely the topology-transparent schedule from [9], which was found advantageous over using other topology-transparent schedules such as conventional TDMA. IV. S IMULATION R ESULTS In this section we report simulation results to illustrate the performance of the proposed R-BSP scheduling with regards to per-node throughput and scheduling delay. In particular, we present results for the R-BSP schedule (10) and compare it with results for the optimized topology-dependent BSP (TBSP) schedule from [7] and the conventional TDMA schedule
1
TDMA T−BSP R−BSP
Empirical Prob(ρthrough ≥ x)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
x
Fig. 3: C-CDF of ρthrough from (3). N = 8. 1
TDMA T−BSP R−BSP
Empirical Prob(ρthrough ≥ x)
0.9 0.8
V. C ONCLUSIONS
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.04
fixed. By applying spatial reuse, significant throughput gain is achieved in all topology configurations using R-BSP and TBSP compared to the TDMA. Comparing the throughputs of R-BSP and T-BSP we observe some performance advantages for R-BSP. This is due to the frame-length minimization in T-BSP (see [7]), which in turn leads to the scheduling delay advantages shown in Figure 2. The true benefit of R-BSP is highlighted in Figure 4 which shows throughput performance when topology varies. It can be seen that throughput of T-BSP dramatically decreases compared to the static-topology case, while a negligible performance degradation is observed for RBSP. This result demonstrates the effectiveness of the proposed combination of topology-transparent and topology-dependent scheduling for R-BSP, to cope with uncertain topology information. R-BSP is thus able to overcome one of the main limitations of spatial-reuse scheduling (e.g., T-BSP), namely sensitivity to outdated topology information.
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
x
Fig. 4: C-CDF of ρthrough for time-varying topologies. N = 8.
In this paper we have addressed the problem of resource allocation in UWAC networks for time-varying topologies. We have presented a new spatial-reuse schedule that combines topology-transparent with topology-dependent scheduling to achieve robustness to inaccurate topology information. By means of simulation results, we have demonstrate that compared with an optimized topology-dependent schedule, at a slight cost in transmission delay, robustness to topology variations is significantly improved. R EFERENCES
without spatial-reuse. We require a minimal number of original packets per frame of gi = 1 ∀i [see (2)] and assume that minimal hop-distance routing is applied. In the simulations, the R-BSP was solved for a randomly generated set of 100, 000 connected undirected random graphs with N = 8 drawn from Bernoulli random graphs with edge probability of 21 . We measured the network performance for a fixed time of T = 1000 time slots. The effect of topology variations on system performance was investigated for each topology by changing the connection between a single pair of vertices in the connectivity matrix (connected or disconnected) in a period of 2N time slots and repeating the process for all possible one-link variations. In Figure 2 we present the empirical cumulative density function (CDF) for the scheduling delay (4) when the topology is time-invariant. Since scheduling delay is strongly linked to the number of time slots a node is assigned to, a significantly lower delay is obtained for the T-BSP and R-BSP schedules compared to TDMA. Since the frame length is directly minimized in the T-BSP schedule [7], it achieves a slightly lower scheduling delay in almost all topologies than R-BSP. We note that scheduling delay is hardly affected by outdated topology information since delay is measured only for received packets and retransmission is not applied. In Figure 3 we show the empirical C-CDF of the measured per-node throughput (3) when the network topology is
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