1

Achieving 100% Throughput in WDM-PON under the SUCCESS-HPON Architecture Jung Woo Lee*, David Gutierrez*, Kyeong Soo Kim** and Leonid G. Kazovsky* *Photonics & Networking Research Laboratory, Stanford University, Stanford CA 94305 {jungwoo.lee,degm,kazovsky}@stanford.edu **Advanced System Technology, STMicroelectronics, San Jose CA 95131 [email protected] Abstract— In this paper we study the tunable resources scheduling problem for WDM-PON under the Stanford University aCCESS Hybrid WDM/TDM PON (SUCCESS-HPON) architecture, a next generation optical access network. In SUCCESS-HPON a few tunable transmitters and receivers at the OLT are shared by all the users for both downstream and upstream transmission using a centralized light sources approach. Each ONU is assigned a single wavelength for both downstream and upstream data transmission; for the latter, a continuous wave is provided by the OLT for the ONUs to amplitude-modulate upstream data onto it and send it back to the OLT. Through the use of novel scheduling algorithms, it is possible to provide service to all users in the network with just a few tunable transmitters and receivers, making the network very cost-efficient. The performance of these algorithms has been extensively studied through simulations. In this paper we attempt to prove 100% throughput guarantee on a particular scheduling algorithm. We show that the Maximum Weight Matching (MWM) algorithm for admissible traffic with the Strong Law of Large Numbers property is stable and can guarantee 100% throughput in a simplified model of SUCCESS-HPON. We derive this result by converting the scheduling problem under consideration to a generalization of the well-known crossbar input-queued switch scheduling problem, and then use a fluid model of a discrete time switch together with the extended Birkhoff-von Neumann (BvN) Decomposition Theorem. The MWM algorithm can be easily implemented on the SUCCESS-HPON architecture since the number of wavelengths, which determines the complexity of implementation of MWM algorithm, is usually small. This proof of MWM in SUCCESS-HPON is meaningful in that it suggests a practical scheduling algorithm with 100% throughput guarantee as well as determines a theoretical bound. As a byproduct of this research, we prove the extended BvN Decomposition Theorem, which may be useful in proving 100% throughput guarantee of MWM algorithm in resource scheduling (allocation) problems where certain number of users share the same kind of resources, for example, as in the case that K users are sharing N type-A resources and M type-B resources.

I. INTRODUCTION

S

cheduling packets under the constraints of shared resources is critical for the next-generation wavelength-routed Passive Optical Networks (PONs) where tunable transmitters and tunable or fixed receivers are shared by many users in order to reduce the high cost of Wavelength Division Multiplexing (WDM) optical components. The scheduling problem we study in this paper is for WDM-PON under the Stanford University aCCESS - Hybrid WDM/TDM PON (SUCCESS-HPON) This work was sponsored in part by the Stanford Networking Research Center (SNRC, http://snrc.stanford.edu).

architecture, a next-generation optical access architecture [1].1 The analysis of scheduling algorithms for this architecture is challenging mainly due to the unique features of WDM-PON under SUCCESS-HPON. Currently a few of these algorithms have been proposed and their performance has been evaluated through comprehensive simulations, including the Batch Earliest Departure First (BEDF) and Sequential Scheduling with Schedule-Time Framing (S3F) algorithms [2]. However an analysis of these algorithms has not been made so far to prove that they provide 100% throughput guarantee for admissible input traffic. In this paper, we study the stability property of a scheduling algorithm for general admissible traffic patterns and show that Maximum Weight Matching (MWM) for admissible traffic with the Strong Law of Large Numbers (SLLNs) property is stable under the following assumptions: (1) the size of the messages is fixed and (2) the propagation delay between Central Office (CO) and the Optical Network Units (ONUs) is the same for all ONUs. Note that the first assumption is actually the case for systems employing fixed-size slots internally with a Segmentation and Reassembly (SAR) sublayer to accommodate variable-size message formats like Ethernet frames or IP packets. As for the second assumption, we can also imagine practical deployment scenarios even though the distance between CO and ONUs varies, equal-length distribution fibers, cut and prepared in advance by service providers, are used; in this case, any unusued portion of the fibers are stored inside ONUs equipped with fiber cassettes [3].2 We obtain the 100% throughput guarantee result by first formulating the scheduling problem for WDM-PON under SUCCESS-HPON as a generalization of the well-known crossbar switch scheduling problem [4]; then we apply a fluid model for discrete time switches with the use of the extended Birhoff-von Neumann (BvN) Decomposition Theorem, also proved in this paper. The structure of this paper is as follows: In section II, we briefly describe the SUCCESS-HPON architecture; in Section III, we convert its scheduling problem into an equivalent well-known crossbar input-queued switch scheduling problem; in section IV, we extend the BvN theorem, which is necessary for the proof of stability of MWM scheduling; in section V we prove MWM is stable for admissible 1

We have changed the name of the architecture from “SUCCESS” to “SUCCESS-HPON” to distinguish it from other architectures under the same SUCCESS initiative at PNRL in Stanford University. 2 We are currently working on generalizing the results obtained in this paper by relaxing both assumptions.

2 traffic with SLLNs property; section VI concludes this paper. λ 3, λ 4, … Central Office

λ 1, λ 2

λ *3, λ 4, …

λ *1, λ 2 λ *1 RN

U D

U D

U D

RN 1

λ 41

λ4

λ1

OLT

λ *3 λ3

RN

λ 42

T1

λ 43

λ2

K

2

TM

R1

RN

Figure 2. OLT Block Diagram

RN

λ 21

TDM-PON ONU RN TDM-PON RN

λ 22

λ 23

WDM-PON ONU RN WDM-PON RN

Figure 1. SUCCESS-HPON Architecture.

II. SUCCESS-HPON ARCHITECTURE Fig. 1 illustrates the SUCCESS-HPON overall architecture. A single-fiber collector ring with stars attached to it formulates the basic topology. The collector ring strings up Remote Nodes (RNs), which are the centers of the stars. The ONUs attached to the RNs on west side of the ring talk and listen to the transceiver on the west side of the Optical Line Terminal (OLT) at the central office, and likewise for the ONUs attached on the east side. Logically, there is a point-to-point connection between each WDM-PON ONU and OLT. No wavelength is reused on the collector ring. When there is a fiber cut, the affected RN will sense the signal loss and flip their orientation. The SUCCESS-HPON architecture has unique features that have direct impact on the design of its scheduling algorithms: (1) tunable transmitters and receivers at the OLT are shared by all ONUs on the network to reduce transceiver counts; and (2) the tunable transmitters at the OLT not only generate downstream data traffic but also provide the ONUs with optical Continuous Wave (CW) bursts for their upstream transmission, which eliminates the need of expensive DWDM sources at ONUs. Therefore a scheduling algorithm for WDM-PON under SUCCESS-HPON has to keep track of the status of all shared resources (i.e., tunable transmitters, tunable receivers and wavelengths assigned to ONUs) and arrange them properly in both time and wavelength domains. For example, if an ONU has requested to send upstream data to the OLT, one tunable transmitter to send the CW, one tunable receiver to receive the upstream data at the OLT, and one particular channel need to be reserved. For downstream data from the OLT to an ONU only the reservation of a tunable transmitter at OLT and the channel between the OLT and ONU is necessary. Because the tunable transmitters are used for both upstream and downstream traffic but tunable receivers are for only upstream traffic, we usually need more transmitters than receivers, i.e., K ≥ M ≥ N where K, M and N are the number of channels (or wavelengths), the number of tunable transmitters, and the number of tunable receivers respectively.

III. THE SCHEDULING PROBLEM AS A GENERALIZATION OF THE WELL-KNOWN CROSSBAR SWITCH SCHEDULING PROBLEM Fig. 2. and Fig. 3. show the logical block diagram of the OLT and its equivalent K x M crossbar input-queued switch, respectively. We consider a SUCCESS-HPON3 network with K ONUs (therefore K wavelengths), M tunable transmitters, and N tunable receivers. As previously mentioned, we assume that there are more tunable transmitters than tunable receivers, i.e., K ≥ M ≥ N. The SUCCESS-HPON OLT polls to check the amount of upstream traffic stored inside ONUs and sends grants with optical CW bursts to allow ONUs to transmit the queued upstream traffic. Since there is neither separate control channel nor control message embedding scheme using escape sequences as in [5], the SUCCESS-HPON MAC protocol employs in-band signaling, i.e., the ONU reports the amount of upstream traffic waiting on a field of the upstream data frame.

Channels 1

U D

T•R

1

T•R

N

1

U D

T

T

U K

M

D

Figure 3. Equivalent K x M Crossbar Input-Queued Switch.

As shown in Fig. 3, the equivalent K x M crossbar input-queued switch has K input nodes which correspond to K ONUs or channels. Upstream transmission requests from channel i are stored as a packet at the input of a FIFO virtual output queue (VOQ) buffer, denoted here by VOQiu. This virtualization avoids the loss of throughput that the head-of-line blocking might cause. Similarly, downstream transmission requests to channel j are stored as a packet in VOQjd . The size of the packet is equal to the size of the data reported by an OLT or ONUs. The output nodes of the switch represent shared resources, i.e., tunable transmitters and tunable receivers. As shown in Fig. 3., each node in the top N nodes has one tunable transmitter and one tunable receiver. These nodes can be thought of as the output ports to which an upstream request should be connected to for scheduling an upstream data 3

From now on in this paper we’ll use the term “SUCCESS-HPON” to refer to “WDM-PON under the SUCCESS-HPON architecture”.

3 transmission. This is because for upstream data both a tunable receiver and a tunable transmitter need to be reserved. The other M - N nodes have a tunable transmitter only; downstream request can be connected to one of these output nodes as well as to the top N nodes for scheduling a downstream data transmission. In a conventional crossbar input-queued switch, a packet is physically ‘transferred’ from input i to output j when input node i is connected output node j after a scheduler’s decision. However in our equivalent crossbar switch model, to transfer means to reserve. In other words when input i and output j are connected for an upstream data request, it means that a tunable transmitter and a tunable receiver in output j are reserved for an upstream data request from ONU i and the transmission time of packets corresponds to the duration of the reservation. Since we are assuming fixed size cells transmission time is constant for all packes. Therefore, while a packet with size t is being transferred from input i to output j (i.e., a request from i is reserving a tunable transmitter or a tunable transmitter and a tunable receiver for the case of upstream) any other packet from input i can not be scheduled to any output and a packet cannot be scheduled to output j during t ( i.e., channel i cannot be used for other transmission requests and resource previously reserved cannot be used during t.). Also note that when we schedule a tunable receiver we can reschedule this receiver in next time slot since the actual time of the receiver usages is delayed by same amount of time in all cases due to our fixed length packet and constant delay assumptions. Now we just formulated the scheduling problem in the SUCCESS-HPON OLT as a generalization of the well-known crossbar switch scheduling problem. Before we prove MWM is stable for admissible traffic with SLLNs property, we extend the BvN theorem which will be used in the proof.

as: 1 - T(k, m, n) = { A ∈ (k X 2 ) matrices | k ≥ m ≥ n, aij ∈{0,1}, ai1 + ai2 ≤ 1 for all i,

k

∑a

i1

≤ n, and

i =1

∑a

ij

≤ m}

i, j

1-T(k, m, n) is therefore a subset of T(k, m, n) where each element is 0 or 1. Now we are ready to state the extended Birkhoff-von Neumann Theorem. Theorem 1: Extended BvN Decomposition Theorem: For a matrix A ∈ T(k, m, n), there exists a set of non negative numbers αk and Pk∈ 1- T(k, m, n) such that A ≤ ∑ α k P k and ∑ α k = 1. k

k

To prove this, we define following transformation: Definition 3: Transformation Tf maps a matrix A = {aij} ∈ T(k, m, n) onto a matrix A’ = {aij’} ∈ (K x K) doubly sub- stochastic matrix. The transformation steps are as follows: Step 0 : j ← 1 (Starts from the first row of the matrix.) K

Step 1: If

∑a

i1

≤ 1, then aj 2 ← ( aj 2 - t), aj1 ← ( aj1 + t) and j ← (j + 1)

i =1

k

where t = min(aj 2, 1 - ∑ ai 1) i =1

Repeat Step 1 until j becomes K. Step 2 : Construct a K x K matrix A' = { ai j '} from the matrix A :  ai j Initially, ai j ' =   0 Step 3 : j ← 1 and m ← 2 K

Step 4 : If

∑a

im

( j = 1, 2 )   (otherwise) 

≥ 1, then aj m ← ( aj m - t ), aj m + 1 ← t and j ← ( j + 1 )

i =1

k

where t = min ( aj m, ∑ ai m - 1 ). Repeat Step 4 until j becomes K. i =1

IV. EXTENSION ON BIRKHOFF-VON NEUMANN THEOREM In this section we briefly review the Birhoff-von Neumann Theorem and extend it to a broader classes of matrices. Birkhoff-von Neumann Decomposition Theorem: For a doubly (sub-) stochastic matrix Λ’, there exists a set of positive numbers αk and permutation matrices Pk such that Λ ' ≤ ∑α k P k and k

∑α

k

=1

k

The equality holds for a doubly stochastic matrix. As it will be explained in the next section, a traffic load matrix under the SUCCESS-HPON architecture is generally not in the form of doubly stochastic matrix or doubly sub-stochastic matrix. Therefore it is necessary that we extend the theorem to the class of matrix to which our traffic load matrices belong to. Definition 1: A subset of K x 2 matrices, T(k, m, n), is defined as: T(k, m, n) = { A ∈ (k X 2 ) matrices | k ≥ m ≥ n, aij ∈ non - negative real, ai1 + ai2 ≤ 1 for all i,

k

∑a

i1

i =1

≤ n, and

∑a

i, j

≤ m}

i, j

T(k, m, n) could be considered as the generalized concept of doubly sub-stochastic matrix. Definition 2: A subset of K x 2 matrices, 1-T(k, m, n), is defined

Step 5 : m ← m + 1 and repeat Step 4 and Step 5 until m becomes K - 1.

Note that the resulting matrix A’ is doubly sub-stochastic. Definition 4: The inverse transformation Tf-1 is defined as usual, i.e., it maps A’ onto A via inverse transformation steps. Now we are ready to prove the Extended Birkhoff-von Neumann Theorem. Proof of the Extended Birkhoff-von Neumann Theorem: Given a matrix A = {aij} ∈ T(k, m, n), we transform A using Tf to get A’. Since A’ is a doubly sub-stochastic matrix, by the BvN theorem we get A' ≤ ∑ α k P k and ∑ α k = 1. k

k

where Pks are permutation matrices. Now we take the inverse transformation Tf -1 of both sides. Due to the linearity of the A ≤ ∑ α k Tf k

−1

( P k ) and

∑α

k

= 1.

k

inverse transformation Tf -1 , we have It can be shown that Tf -1( Pk) belongs to 1-T(k, m, n). This completes the proof. Fig. 4. show an example of decomposition by the extended Birkhoff-von Neumann theorem.

4 V. PROOF OF 100% THROUGHPUT GUARANTEE OF THE MWM ALGORITHM In this section we describe the MWM algorithm for SUCCESS-HPON and provide a proof of its stability for admissible traffic with SLLN property under the following assumptions: (1) the size of message is fixed and (2) the propagation delays between the OLT and the ONUs are the same for all ONUs. Let Aij(n) denote the number of fixed-size messages or cells that have arrived at VOQij up to time n. We adopt the convention that Aij(0) = 0. We assume that the arrival processes A (n) = [Aij(n)] satisfy the Strong Law of Large Numbers (SLLN), that is for any i = 1,….,K and j = u (for upstream), d (for downstream) with probability one,

lim

n →∞

Aij(n) = λij . n

(1)

We call λij the arrival rate at VOQij.We represent a matching by a K x 2 matrix π = [πij] where if an upstream request of channel i is granted for scheduling, we have πiu= 1, otherwise πiu = 0 and similarly if a downstream request of channel i is granted for scheduling, we have πid = 1, otherwise πid = 0. Definition 5: The arrival process with arrival rate matrix Λ = [λij ], defined to be “admissible” iff (1) holds and no resource is overloaded, in other words,

λiu + λid ≤ 1 for all i,

K

∑λ i =1

K

iu

≤ N, and ∑ λiu + λid ≤ M

(2)

i =1

∑α

k

k

≤ 1.

(3)

k

Let Dij(n) indicate the number of departures from VOQij up to time n. Again assume Dij(0) = 0 and D(n) = [Dij(n)]. Definition 6: A switch operating under a matching algorithm is called “stable” (rate stable) if, with probability one, Dij(n) = λij lim n →∞ n

∀ i = 1,...K and j = u, d

(4)

for any admissible arrival process A(n) = [Aij(n)] with rate λij. Let Zij(n) denote the number of cells in VOQij at time n, including any arrival at time n, then the matrix Z(n) = [Zij(n)] shows the queue occupancy at time n. For any matching matrix π∈ Π, the “weight” fπ(n) of the matching at time n is defined as, fπ (n) = < π , Z(n)> (5) where A, B =

∑A

i, j

Bi, j

A. Maximum Weight Matching (MWM) Algorithm At each time slot, the MWM algorithm will find the matching with the maximum weight among all possible matchings in Π. If there is more than one matching with the maximum weight, one of them is chosen randomly. We denote the maximum weight matching and its corresponding weight at time n by πw(n) and fw(n) respectively. That is, (6) π w(n) = arg max π {f π (n)} f w(n) = <πw(n), Z(n)>.

(7)

B. Proof In [6], the fluid model for a discrete-time switch was introduced. We will use this fluid model with its proofs and justifications in this paper. Theorem 2: Maximum Weight Matching (MWM) for admissible traffic with Strong Law of Large Numbers (SLLNs) property is stable in SUCCESS-HPON. Proof: Let Tπm be the cumulative amount of time that matching π is used under scheduling scheme m up to time n. For a discrete-time switch the following three equations govern the dynamics of the system: (8) Zij (n) = Zij (0) + Aij (n) − Dij(n) Tπm (⋅) non - decreasing, and ∑ Tπm (n) = n π ∈Π

n

where K, M and N are the number of channels (or wavelengths), the number of tunable transmitters, and the number of tunable receivers respectively as before. Note that any admissible arrival rate matrix belongs to T(k, m, n) and any valid matching matrix belongs to 1- T(k, m, n). Therefore, from the expended BvN theorem, for a given admissible arrival rate matrix Λ∈ T(k, m, n), there exists a set of non negative real numbers αk and Pk∈ 1- T(k, m, n) such that

Λ = ∑α k P k and

for two matrices A and B of the same size.

Dij(n) = ∑∑ πij 1{ Zij(l) > 0 } (Tπm (l ) − Tπm (l −1))

(9) (10)

π ∈Π l =1

Equation (8) states that the number of cells in VOQij is equal to number of arrivals minus the number of departures. The next equation shows the relationship between number of departures, Dij(n) and the cumulative time of the matching π, Tπm . As shown in [6], the continuous equations governing the dynamics of the fluid model of the system described above are as follows:

Zij (t ) = Zij (0) + λijt − Dij(t)

(11)

Tπm (⋅) non - decreasing, and ∑ Tπm (t ) = t

(12)

⋅

⋅

π ∈Π

Dij(t) = ∑ πij Tπm (t ) 1{ Zij(t) > 0 } π ∈Π

(13)

Let Z(t) and D(t) be Z(t) = [Zij(t)] and D(t) = [Dij(t)]. Now consider the quadratic Lyapunov fuction L(t) = and consider all t such that the fluid quantities are differentiable and hence L(t) is well defined. By definition, ∂ ∂ L(t) = 2 < Z (t ), Z (t ) > ∂t ∂t ∂ ∂ = 2 < A(t), Z (t ) > − 2 < D(t), Z (t ) > ∂t ∂t ∂ = 2 < λ , Z(t) > − 2 < D(t), Z (t ) > . ∂t

(14)

5

Figure 4. Decomposition Example: A →T(4,3,1)

For any admissible load matrix Λ, < λ , Z(t) > ≤ < ∑ α k P k , Z(t) > where ∑ α k ≤ 1 k

(15)

k

( from the extended BvN theorem) ≤ ∑ α k < P k , Z(t) > k

≤ ∑α k f w k

≤ f w.

Also we have <

∂ ∂ D(t), Z (t ) > = ∑ f w (t) ⋅ Tπw (t) t ∂t ∂ π ∈Π ∂ = f w (t) ⋅ ∑ Tπw (t) ∂ π ∈Π t

REFERENCES (16)

= f w (t). By applying (15) and (16) into (14), we get

∂ ∂ L(t) = 2 < λ , Z(t) > − 2 < D(t), Z (t ) > ≤ 0 ∂t ∂t

The MWM algorithm could be easily implemented on the SUCCESS-HPON architecture since the number of wavelengths, which determines the complexity of implementation of MWM algorithm, is usually small. The stability proof of this paper suggests a practical scheduling algorithm with 100% throughput guarantee for WDM-PON under the SUCCESS-HPON architecture and provides a maximum throughput theoretical bound. As a byproduct of this research, we proved the extended BvN Decomposition Theorem, which is believed to be useful in proving 100% throughput guarantee of MWM algorithm in resource allocation problems where certain number of users share same kind of resources (e.g., when K users are sharing N type-A resources and M type-B resources).

[1]

[2]

(17)

[3]

This completes the proof. [4]

VI. CONCLUSION In this paper, we show that Maximum Weight Matching (MWM) algorithm for admissible traffic with the Strong Law of Large Numbers (SLLNs) property is stable in a simplified model of WDM-PON under the SUCCESS-HPON architecture. We derive this result by first converting the scheduling problem being considered to a crossbar input-queued switch scheduling problem, and then applying the fluid model of a discrete time switch with the use of the extended Birhoff-von Neumann (BvN) Decomposition Theorem that has also been proved in this paper.

[5]

[6]

F-T. An, K.S. Kim, D. Gutierrez, S. Yam, E. Hu, K. Shrikhande, and L. G. Kazovsky, “SUCCESS: A next generation hybrid WDM/TDM optical access network architecture”, IEEE/OSA Journal of Lightwave Technology, vol. 22, no. 11, pp. 2557-2569, Nov. 2004. K. S. Kim, D. Gutierrez, F.-T. An and L.G. Kazovsky, “Design and performance analysis of scheduling algorithms for WDM-PON under SUCCESS-HPON architecture”, IEEE/OSA Journal of Lightwave Technology, vol. 23, no. 11, pp. 3716-3731, Nov. 2005. K. S. Kim, “Past, present, and future of fiber-to-the-home solutions,” EE201A Seminar, Stanford University, Dec. 3, 2001 (http://www.stanford.edu/~kks/Papers/ftth.pdf). N. McKeown, V. Anantharam, J. Walrand, “Achieving 100% throughput in an input-queued switch”, Proc. of INFOCOM 1996, Mar. 1996. G. Kramer, B. Mukherjee, and G. Pesavento, “IPACT a dynamic protocol for an Ethernet PON (EPON),” IEEE Commun. Mag., vol. 40, pp. 74-80, Feb. 2002. J.G. Dai and B. Prabhakar, “The throughput of data switches with and without speedup,” Proc. of INFOCOM 2000, pp. 556-564.