DELAY ANALYSIS AND QUEUE-LENGTH DISTRIBUTION OF A SLOTTED METRO NETWORK USING EMBEDDED DTMC

Tülin ATMACA

Institut Telecom/Telecom SudParis– 9 rue Charles Fourrier, 91011, Evry, France

tulin.atmaca@it-sudparis.eu

Tuan Dung NGUYEN

Institut Telecom/Telecom SudParis– 9 rue Charles Fourrier, 91011, Evry, France

dung.nguyen@it-sudparis.eu

Abstract - In this paper, we investigate performance analysis on queuing delay and queue length distribution of a slotted bus-based optical metropolitan area network (MAN) using fixed length packet format and a single wavelength. The studied network is a unidirectional bus on which many access nodes are connected to transmit fixed length packets to a centralized Hub node over a single wavelength. Each access node is modeled by an embedded discrete time Markov chain (EDTMC). The solution of this EDTMC allows computing the approximate probability that the next downstream access node on the bus sees the slot free. By using the recurrent analysis technique, we outline an approximate queue-length distribution as well as an approximate mean waiting time for each access node. The analytical results are finally compared with the simulation results. We have seen that the results strongly depend on the typical number of access nodes in the network. Both simulation and analytic models capture the expected behavior of mean waiting time and queue-length distribution but the difference between two models becomes more important as the transmission priority of ring node decreases. Keywords: MAN, slotted OPS bus, performance analysis, queueing delay, queue length distribution, embedded discrete time Markov chain.

I. INTRODUCTION The growth of video and interactive data traffic today leads to the introduction of high-speed optical networks at the access and metro areas. Among many technologies, the optical packet switching (OPS) network is gaining high attention of service providers thanks to its scalability and performances compared to existing technologies such as SONET/SDH. Generally, a MAN uses ring or bus topologies (i.e DBORN [1]), in which a master (Hub) node communicates with access nodes in point-to-multipoint mode, and access nodes communicate with the master node in multipoint-to-point mode. A major problem in such architectures is how to efficiently control the access to transmission resources for the upstream communication, where many access nodes are in competition for sending packets to the Hub node. The downstream communication does not pose a problem since it is a point-to-multipoint communication (broadcast or diffusion) from the Hub node to access nodes. We consider in this work a slotted bus-based optical MAN, and focus specially on the upstream path. The upstream bus is a unidirectional bus connecting a number of access nodes to the Hub. The upstream bus consists of two wavelengths: one

for the data transmission, the other for the synchronization and control purpose. The bus is slotted into fixed length slots which are equivalent to the packet length and an access node can insert a fixed length packet at the beginning of each time slot. Downstream access nodes can only insert optical packets into empty slots left by upstream access nodes. In such a typical bus-based network, it is well known that the performance of an access node strongly depends on its position on the bus due to the transmitting correlation with other upstream nodes. The mutual correlation among bus nodes makes the performance analysis of such types of network difficult. For the asynchronous bus-based network, there have been a number of works [2], [3] that approximately computed the queue length distribution (and hence the mean queue length) of each access node on the bus. To our knowledge, there have been also some works that specifically investigated the analytical model of the synchronous bus-based network with passive optical components such as [4], [7]. Regarding M/G/1 priority queuing systems, [4] following Jaiswal [5] and Takagi [6] are commonly derived the mean waiting time at each ring node while [7] uses numerical methods to obtain the stationary probability of the free slot in the transit line before computing the mean waiting time but obviously no works approximately computed the queue length distribution in the synchronous network. We propose in this paper an analytical model for performance analysis of the upstream bus of slotted bus-based optical MAN. Each access node is modelled by an embedded discrete time Markov chain (EDTMC). The solution of this EDTMC allows us to compute the approximate probability that the next downstream node on the bus sees the slot free. By using the recurrent analysis technique, we are able to compute an approximate queue-length distribution as well as an approximate mean waiting time for each access node. The mean waiting time at a node is defined as the expected time elapsed from the moment when an optical packet arrives at the local buffer of the node until its transmission begins. The rest of this paper is organized as follows. We first describe the network architecture and our analytical model. Next, we present the node-by-node analysis technique, thanks to the free slot probability, to outline an approximate queue-length distribution as well as an approximate formula calculating the mean waiting time in local buffer of each access node. The numerical results are then provided and compared with those of simulation. Finally we give some conclusions and discussions of future works.

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II. NETWORK ARCHITECTURE

In this work, we study and analyze the performance of access nodes on the upstream bus of an optical MAN that provides a shared transmission medium for carrying optical packets from several access nodes to the point of presence node – Hub (Figure 1).

Figure 1 – Upstream bus logical architecture of a slotted bus-based MAN

Each access node has two separated planes: a control plane treating incoming control packets in electronic domain and a data plane processing incoming optical packets in optical domain. In order to reduce network cost, we use passive optical component in the data plane, in the sense that the incoming transit optical packets pass transparently without being converted to electronic signal or dropped by intermediate nodes. With this concept, we reduce the number of receivers used at each node for the upstream communication. We need only one receiver at each node to receive optical packets sent by the Hub node for the downstream communication. Therefore, there is no direct communication between access nodes in this architecture. Any communication between two nodes is done through the Hub node (the so-called “hub and spoke” architecture). We recall that the downstream communication is out of the topic treated in this work. The wavelengths (control and data) are slotted into fixed length slots in time. As shown in Figure 1, each control packet circulating on the control wavelength provides information about the respective time slot on the data wavelength. In this work, we simplify the network architecture by supposing that a control packet simply allows the node to detect whether the corresponding data slot is free or busy (i.e., occupied by an optical packet sent by an upstream node). It is worth noticing that the use of control packets would become more useful in case where we want to introduce active functionalities into the node architecture, namely the priority scheduling between transit and local packets (i.e., transit packets must be also treated), based on the packet identifier contained in control packets. To have sufficient time for the processing of control

packet, optical transit packet is delayed in the Fibre Delay Line (FDL) of the node. FDL length is equivalent to timeslot. The functionality of an access node is described as follows. The local traffic coming from client networks is aggregated into common electronic buffer of the node. The node creates optical packets (whose length is equal to the time slot length) from client electronic packets in the common electronic buffer. The study of optical packet creation mechanisms forms another complex problem and is not covered in this work, but it is investigated in our previous works [8]. The created optical packets are stored in local electronic buffer (usually fast random access memory - RAM) waiting for the transmission on the optical wavelength. At the same time, the access node reads control packets (sending by upstream nodes) on the control wavelength, seeking for free data slots on the data wavelength. Since this is a slotted (synchronous) network, the node can insert an optical packet (if any) only at the beginning of a free data slot, while updating the content of the corresponding control packet of this data slot. We can notice that in this network, as well as in any typical bus-based network, the first upstream node that begins the transmission on the shared data wavelength benefits from the total free bandwidth, hence having no problem for transmission (position priority). However, the transmission of other downstream nodes depends on the distribution of free data slots left by upstream nodes. In other words, downstream nodes performance is correlated to the packets sent by upstream nodes.

III. ANALYTICAL MODEL

We consider an optical MAN with N access nodes (1...N) in which node 1 has the most position priority to send their packets) sharing one wavelength operating at 10 Gbps. We assume that each node has an infinite local buffer, and client packets stored in this buffers. The optical packets are created by assembling client packets waiting in this buffer according to the packet creation mechanisms chosen. So, optical packets are serviced in First-Come-First-Serve (FCFS) manner. We also assume that optical packets at node i arrive according to a Poisson process with rate )(iλ , and that their service time is deterministic (i.e., optical packet has fixed length equal to the slot length T). As mentioned above, we approximately model each node by an EDTMC. To be able to solve the chain, we need to know the probability that the node sees the current data slot free which is usually an unknown parameter. Fortunately, for the node 1, the probability that it sees the data slot free is always equal to 1. Thus, from the solution of node 1, we are able to compute the probability that the next downstream node sees the data slot free, and so on. To obtain a tractable model for the network considered, we are interested in particular moments of the evolution of the state of the system. Instead of analyzing the state of the system at continuous time, we analyze the state of the system at discrete moments tk that are the beginning of each data slot k,

Ctr i+1

Hub

Ctr 1

Slot 1

Ctr 2

Slot 2

Ctr i

Slot i Slot i+1

Data wavelength

Control wavelength

Control packets

Data packets

Control

Unit

Electronic buffers

Add traffic

Optical Transparent transit path

Control wavelength Optical

fiber

Data wavelength FDL

Node Node Node

(First Node) (Last Node)

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k = 1, 2, 3..., with tk+1 = tk + T where T is the slot length. This consideration brings the system back to a simple description of state for studying. At the beginning of each data slot, supposing that the precedent service was finished, we have in the following situations:

• A new optical packet is transmitted if the data slot is free. • If the data slot is busy, all optical packets must wait in the

local buffer until detection of next free data slot.

Therefore, at these moments, the state of the system is completely defined by the number of optical packets in the buffer and the slot’s state (free-busy). Let Xk, k = 1, 2, 3…, be the number of local optical packets in the buffer at tk. Let pk, k = 1, 2, 3…, be the state of data slot k at tk which take values 0 if the slot is busy, 1 otherwise. The couple (Xk, pk) is enough to predict the later evolution of the system. However, this description has not the Markovian property because between two consecutive moments (tk, tk+1), the evolution of the number of clients Xk (i.e., optical packets) in the system depends on the Poisson arrival of optical packets, but also on the unknown arrival process of transit traffic coming from upstream nodes and as well as on pk . For node 1, which is the first upstream node beginning the transmission on the shared wavelength, pk is equal to 1 for any value of tk. Thus, the node 1 is completely characterized by the Markovian process {Xk}.

Figure 2 – The evolution of Xk, k = 1, 2, …

For nodes i > 1, since we do not know the process {pk}, supposing that we only know the mean value p of the process {pk} at steady state (the computation of p is given in next sections). Therefore, to obtain a tractable model, we suppose that at any moment tk, the data slot k is free with probability p independent of k, and busy with probability 1-p. Replacing {pk} by its mean p in the description of the node for all discrete time tk, we obtain the process {Xk, p}, which, for the sake of simplicity, can be noted as {Xk}. It is clear that {Xk} forms a discrete time Markov chain (Figure 2) which represents queue-length evolution in function of time. Let pts(n)(k) denotes the probability of having n packets in the local buffer at tk.: pts(n)(k) = P[Xk = n]. We need to know the stationary probabilities ∞→klim {pts(n)(k)} in order to compute the waiting time of local optical packets. A. Mean waiting time computation

Let consider a node i, i=1,2,3…, which sees a data slot free with probability p(i) (or busy with probability 1-p(i)) at the

beginning of each slot. A slot is busy if it is occupied by an optical packet transmitted by an upstream node. If we consider K time slots, K ∞, observed at node i, then the number of occupied slots is equal to the number of optical packets transmitted by all upstream nodes j =1,…,i-1 during K time slots. Thus, the probability that a slot is busy seen by node i is easily computed as follows:

∑∑−

=

−

=

==−1

1

)(1

1

)()( **11i

j

ji

j

ji TKK

p ρλ (1)

where T is slot length, Tjj *)()( λρ = is the mean offered load of node j. We have also the following equations for p(i):

p(1) = 1 and )(1

1

)( 1 ji

j

ip ρ∑−

=

−= , i=2,…,N (2)

We may now compute the mean waiting time in the local buffer of node i. Let W be the mean waiting time of a local optical packet, 0W be the mean waiting time of a local optical

packet arriving in an empty buffer, and nW be the mean waiting time of a local optical packet arriving in a non-empty buffer containing n optical packets, we have: nWWW += 0 (3)

Since an optical packet can be inserted only at the beginning of a data slot, if it arrives during a data slot, it must first synchronize to the beginning of the next data slot, then looking for a free data slot for transmission. We can compute

0W as follows:

)(*)0(0 Rs WtpW += (4)

Where )0(p is the probability that the buffer is empty, st is the

mean synchronization time, and RW is the mean searching time a packet takes to find a free data slot and begins its transmission. During a time slot of length T, an optical packet has the same probability to arrive just after the beginning, or at the middle, or at the end of the time slot. Thus st is the mean value of a uniform distribution in interval T: 2/Tts = (5)

RW is easily computed as follows:

Tp

pW i

i

R )(

)(1 −= (6)

Now we come to compute nW . An optical packet, arriving in a non-empty buffer containing n optical packets, must wait for the following services before being transmitted: • A residual service time Rt of the packet being

transmitted: Rt = T/2 (7)

• The service completion time Ct of all n preceding optical packets. Each service completion time is composed of a research time RW for free data slot followed by a constant service of length T:

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Ct = RW + T (8) Therefore we obtain the following equation:

nW =∑∞

=1

*)([n

np ( Rt + RW +n* Ct )] (9)

Using the normalization condition )0(p =1-∑∞

=1

)(n

np and

the Little formula for computing mean waiting time

∑∞

=

=1

)(1

n

nnpWλ

, combined with equations (3) – (9), we are

able to compute the final formula of W :

)(

)(

)(

)(

1

)1(2

i

i

i

i

p

ppTT

Wρ−

−+= (10)

B. Queue-length distribution estimation

Now, in order to estimate the queue-length distribution, we consider an access node i (i ≥ 1) in the network. Since the arrival process is supposed Poisson with the mean rate )(iλ , the probability that c packets arrive during a time slot T at node i

is cα = Tc

ecT λλ −

!)( .

Let denote p(i)(n) the probability of having n optical packets in the local buffer. We decompose, in particular, the empty state of the local buffer (state 0) into two states: state 0* and state 0** as follows:

i. state 0* that there is no departure at the beginning of the data slot;

ii. state 0** that there is a packet just sent at the beginning of the data slot;

iii. now assume that p(i)(0) = p(i)(0*) + p(i)(0**) Note that the probability p(i)(0*) of access node i is equal

to the probability of having free slot for the access node i+1. It means that: p(i)(0*) = p(i+1)

Let us consider that just after data slot (k-1), there are i packets in the local buffer. Just after data slot k, there are i-1 packets in the local buffer if, during a time slot T, no packets arrived and the data slot (k+1) is not occupied by upstream nodes (hence the first optical packet in local buffer will be sent). On other hand, there are i-1+c packets in the local buffer if, during a time slot T, c packets arrived and the following data slot is free, or c-1 packets arrived but the following data slot is occupied. These state evolutions can be represented by the Markov chain in Figure 3.

Figure 3 – EDTMC’s graph of the state i (i >0)

As shown in the figure, we have the probability that the state i returns to the state i-1 is )(

0ipα ; the probability that the

state i returns to itself is )(1

)(0 )1( ii pp αα +− ; and the

probability that the state i comes to the state i+c is )(

1)( )1( i

ci

c pp ++− αα . Let us consider that just after data slot (k-1), only an

optical packet is sent, leaving behind it the empty buffer, (i.e. the state 0**). In this case, we have the probability that the state 0** returns to the state 0* is )(

0ipα . We have also the

probability that the state 0** returns to itself is )(

1)(

0 )1( ii pp αα +− and the probability that the state 0**

comes to the state c is )(1

)( )1( ic

ic pp ++− αα . The EDTMC of

the state 0** is shown as follows:

Figure 4 – The EDTMC’s graph of the state 0**

In particular, we deduce the EDTMC for the state 0* like following:

Figure 5 – EDTMC’s graph of the state 0*

Based on the above described EDTMCs, we have:

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

≥++=

++=+=

≥+−==

∑+

=+−++

−

1

1

'1

)('1

**)('1

*)()(

'0

)('1

**)('1

*)(**)(

'0

**)('0

*)(*)(

)()(1

'

)(00

'

1,)()0()0()(

)1()0()0()0()0()0()0(

1,)1(

k

jjk

ik

ik

ii

iiii

iii

ik

ikk

i

kjpppkp

ppppppp

kppp

ααα

ααααα

ααααα

(11)

Note that all of EDTMCs are irreducible and aperiodic

[13]. Hence, we affirm that the stationary probabilities p(i)(0*), p(i)(0**), p(i)(1), ... exist. By solving (11), we obtain finally:

)()(*)( )0( iii pp ρ−= (12) where )(iρ is the load generated by node i, and

p(i)(0) = '0

)()( /)( αρ iip − (13) From p(i)(0), we are able to deduce easily p(i)(1), p(i)(2),..

As expected, using Little formula with these probabilities, we obtain the same values as (10).

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(*)The offered network load is defined as the ratio of the sum of traffic volume offered to all nodes to the network transmission capacity

Figure 6 – a) The mean waiting time with different offered network loads from node 1 to 9; b) The mean waiting time with different slot lengths for the offered network load of 0.5 from node 1 to 9; c,d,e) The queue-length probability of

Local buffer for three nodes 1,4,9 with the offered network load of 0.5.

IV. NUMERICAL RESULTS We present results obtained by analytical model and by simulation in Figure 6. We use discrete-event network simulation tool [11] to simulate the network with 10 ring nodes (node 10 is the Hub node) transmitting on one wavelength at 10Gbs. Hence, the transmission priority increases from node 1 to node 9. All mean values in our simulation results are computed with an accuracy of no more than a few percents at 95% confidence level. The figure 6a shows the mean response time at each access node for the time slot equal to 1µs as the offered network load(*) increases. We first observe that both

simulation and analytic models capture the expected behavior of mean waiting time but the difference between them increases as the priority of ring nodes decreases. Moreover, the mean waiting time is likely to increase rapidly as the transmission priority decreases or as the offered network load increases. An explanation is that a successful transmission at low priority nodes takes on average a longer time than at higher priority nodes. The number of occupied data slots becomes more and more important leading to excessive waiting time at the lowest nodes. We continue to focus on the analysis of the mean waiting time obtained for the same offered network load of 0.5 but with different slot lengths (Figure 6b). The results

1,00E-07

1,00E-06

1,00E-05

Load=0.45 Load=0.50 Load=0.55 Load=0.60

Simulat ion Analyt ic

a) The mean waiting time (sec) with different offered network loads from node 1 to 9;

1,00E-07

1,00E-06

1,00E-05

1,00E-04

Slot=1e-6(s) Slot=2e-6(s) Slot=5e-6(s) Slot=1e-5(s)

Simulat ion Analyt ic

b) The mean waiting time (sec) with different slot lengths for the offered network load of 0.5 from node

1 to 9;

Q

ueue

-leng

th p

roba

bilit

y (#

optic

al p

acke

t)

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

0 1 2 3 >4

Simulat ion Analyt ic

c)

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

0 1 2 3 >4

Simulat ion Analyt ic

d)

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

0 1 2 3 >4

Simulat ion Analyt ic

e)

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show that the mean waiting time also increases rapidly as the slot length increases. We can see more clearly these results thanks to formula (10) where the mean waiting time is directly calculated by the slot length T. Now, let us consider that the queue-length probability p(i)(n) (n = 1,2,3,…) at access node i in the case of offered network load of 0.5. The figures 6c, 6d, 6e present this probability at three nodes 1, 4, 9. For this experiment, both simulation and analytical models capture the expected behavior of the queue-length probability but the difference between two models increases as the transmission priority decreases or as n increases. We try to explain why there is a high difference between analytical and simulation results for the mean waiting time at last nodes. The principle raison is illustrated in the figure 7.

Figure 7 - Ratio of free slot captured in sample of 200 slots

In this figure, we presents the free slot probability p(i)

(i from 1 to 9) captured in only a sample of 200 slots for the offered load network of 0.50. We note that with the sample of 200 slots, the free slot probability is very variable from stationary probability, especially for the last node. It is clearly shown in the figure that as the lowest node, this probability has the most variation. In other words, these results show that the stationary probability p(i) used for the EDTMC corresponding to the uniform distribution of free slot. But in reality, the free slot observed by access nodes is not uniformly distributed.

V. CONCLUSION AND FUTURE WORK We have studied and analyzed the performance of a slotted bus-based optical MAN with fixed length packet format using discrete time Markov chain. The solution of this EDTMC allowed us, by using the recurrent analysis technique, to compute approximately the mean waiting time and estimate the queue-length distribution at each access node. Also, we have seen that both simulation and

analytic models capture the expected behavior of mean waiting time and queue-length distribution but the difference between two models becomes more important as the node’s position priority decreases (notably at last nodes on the bus-based network) because of non-uniform distribution of free slots viewed in the network simulation. Our future work will include an improved matching of distribution of free slots in transit, as well as a possible extension of our approach to different packet arrival patterns and finite buffer sizes.

ACKNOWLEDGMENT This work was partially supported by the EURO-NF (Network of Excellence)

REFERENCES [1] N.Le Sauze, E. Dotaro, A. Dupas and al., “DBORN: A shared WDM Ethernet Bus Architecture for Optical Packet Metropolitan Network”, Photonic in Switching, July 2002. [2] A. Brandwajn, V.H. Nguyen, T. Atmaca, “A conditional probability approach to performance analysis of optical unslotted bus-based networks”, in Book Current research Progress of Optical Networks, ISBN: 978-1-4020-9888-8, Chapter 4, pp. 65-96, Editor: Maode Ma, February, 2009. [3] H. Castel, M. Chaitou, and G. Hébuterne, “Preemptive Priority Queues for the Performance Evaluation of an Optical MAN Ring”, Proceedings of Performance Modeling and Evaluation of Heterogeneous Networks (Het-Net’05), 2005. [4] G. Hu, C.M. Gauger, and S. Junghans, “Performance Analysis of the CSMA/CA MAC Protocol in the DBORN Optical MAN Network Architecture”, Proceedings of the 19th International Teletraffic Congress (ITC 19), 2005. [5] Jaiswal. Priority Queues. New York: Academic 1966. [6] H. Takagi, Queuing Analysis, Vol.1, pp. 365-373. North-Holland, 1991. [7] T. Eido, F. Pekergin, T. Atmaca, “ Modelling and Performance Evaluation of Improved Access Mechanism in a Novel Multiservice OPS Architecture”, IFIP Networking Conference, Aachen, Germany, 11-16 May, 2009, Springer-Verlag LNCS , ISBN 978-3-642-01398-0, pp. 821-834 [8] Tuan Dung Nguyen, Thierry Eido, Viet Hung Nguyen, Tülin Atmaca, “Impact of Fixed-Size Packet Creation Timer and Packet Format on the Performance of Slotted and Unslotted Bus-Based Optical Man”, International Conference on Digital Telecommunications IEEE ICDT’08, Bucharest, Romania, June 29 - July 5, 2008. [9] G. Hu, C.M. Gauger, and S. Junghans, “Performance Analysis of the CSMA/CA MAC protocol in the DBORN Optical MAN Network Architecture”, Proceedings of the 19th International Teletraffic Congress (ITC 19), 2005. [10] M.H. McDougall, Simulating Computer Systems: Techniques and Tools, The MIT Press. 1987. [11] Network Simulator, http://www.isi.edu/nsnam/ns/ [12] H. Bruneel, B.G. Kim, “Discrete-Time Models for Communication Systems Including ATM”, Kluwer Academic Publishers - 1993 [13] F. David, M. Yilmaz, S. Gjessing, N. Uzun, IEEE 802.17 “Resilient Packet Ring Tutorial”, IEEE Communication Magazine, Vol. 42, No. 3, pp 112 – 118, March 2004. [14] K.V.Shrikhande, and al., “HORNET: A Packet-Over-WDM Multiple Access Metropolitan Area Ring Network”, IEEE Journal on Selected Areas in Communications, vol. 18, No. 10, October 2000.

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