Departure Process Analysis

In a (

ψ

τ

)-Shaper system, bursts are served (transported) by one wavelength and forwarded via the same OLSP. In the analysis, (

ψ

τ

)-Shaper is considered on a

states are the H and L states, which correspond to high and low mean arrival rates, respectively. The MMBP is characterized by four parameters (

α

β

λ

L), where

α

is the probability of changing from state H to L in a slot,

β

is the probability of changing from state L to H in a slot,

λ

_H represents the probability of having a batch arrival at state H, and

λ

L represents the probability of having a batch arrival at state L.

For ease of description, the state change probability is denoted as P_{i j}_, ,

^{i j}

^, ^∈

{ ^{H L}

}

Namely, P_{H L}_, = −1 P_{H H}_, =α and P_{L H}_, = −1 P_{L L}_, =β. The batch sizes at state H and L

possess distributions bH(m) and bL(m), with mean sizes

b

_H and

b

_L, respectively.

Let L represent the mean arrival rate (packets/slot) (i.e., the load), and B the burstiness of the arrival process, it thus have

H H

H L

b b

B L b b

λ λ

β λ α λ

α β α β

⋅ ⋅

= =

⋅ ⋅ + ⋅ ⋅

+ +

. (4)

Figure 13 is drawn in aid of comprehension throughout the analysis. There are five possible events that sequentially occur in a slot as follows: (1) arrival process state change, (2) begin-of-burst departure, (3) packet arrivals, (4) end-of-burst departure, and (5) BATr activation/reset. While Events (1) and (2) occur at the beginning of a slot, Event (3) takes place at any time within a slot, and Events (4) and (5) occur at the end of a slot.

The departure process distribution consists of two parts: burst inter-departure time and burst size distributions. The burst inter-departure time takes values which are integer multiples of a slot. It is defined as the interval from the end of a previous burst to the beginning of the following burst. The goal is to

( )

t

( )s

Legend:

: Imbedded Markov chain epochs;

0 System setting:

ψ = 4;

BoB : Begin-of-Burst;

EoB : End-of-Burst;

find the joint distribution of

t

and , i.e.,

s

P_{t s}_, ( , ), 0, t s t≥ 0≤ ≤s ψ . To approach it, we first obtain the queue length distribution seen by departing bursts, based on an imbedded Markov chain analysis placing the imbedded points at burst departure instants, as shown by the arrows in Figure 13.

Define random variable to be the number of packets left in queue behind by the k

q

th departing burst, say at time slot tk, under the condition that the arrival process is in state yk (=H or L) at tk. Let random variable represent the number of packets that arrive during the burst inter-departure interval, under the condition that the arrival process changes from state y prior to the beginning of the interval, to state z at the end of the interval. Moreover, let random variable denote the number of packets that arrive during the transmission time of an n-packet burst, namely n slots, under the condition that the arrival process changes from state y prior to the beginning of the time interval, to state z at the end of the interval.

uz y

| n

v

z y

In Figure 13, the k^th burst depart at tk, and there are no packet left in the queue. The next packet arrives at t_k +3. BATr is activated and set by

τ

. Since the traffic arrival is under low load, there are not enough packets arrival during t_k +3 and

t

k +3+

τ

. At tk +3+

τ

the BATr is expired, and the (k+1)^st burst starts transmission at the next slot. The burst size is . At the end of the (k+1)

k k

z y

u ₊ ^st burst transmission, there have some packets in the queue. BATr is reset at t_k+1. The (k+2)^nd burst is generated while the queue size is more than

ψ

value. Finally at the end of the (k+2)^nd burst transmitted, since the queue size

q

y⁺₊ is still more than

ψ

value, the (k+3)^rd burst is immediately generated and transmits behind the (k+2)^nd burst.

Accordingly, the next queue length

q

y⁺₊ is determined by the current queue length , number of arrival during the inter-departure time , number of

departure packets, and number of arrival packets during transmission . we find that

q

y u_{z y}_|

| n

v

z y

( ) ^{

¹^|

^}

1 1 1 1

| |

yk zk yk

k k k k k k

k k

y y z y y z

q q u v

^min ^,

qk u ψ

ψ

⁺

+ + + +

+ = + − +

⁺ ⁺

{

, (5)

}

1 1

, , ,

k k k

y y

₊

z

₊ ∈

H L ( )

^a ⁺ ⁼^max

{ }

^a^{, 0}

{

where , and . In Equation (5), a non-negative

term within the parentheses corresponds to the departure of a full-size (=

ψ

) burst;

whereas a negative value corresponds to the departure of a burst due to BATr expiration. Significantly, since BATr is reset or activated after the k^th burst departure time, and and

k k

z y

u ₊ min 1| , }

k yk zk yk

q u

y z

vk1 k1 + ψ +

{

+ + are independent of any events that occur prior to time index k,

{

k^, ^,

}

^, ¹

}

y k

q y ∈ H L k≥ is hence an imbedded Markov chain.

Based on Equation (5), we can derive the limiting distributions of the queue length seen by departing bursts, rather than at all points in time. Notice that fortunately, such distribution is sufficient enough to determine the departure process distribution. Before we proceed, let us first derive the distribution for the number of packets that arrive in any given interval. Let denote the probability that m packets have arrived in an interval of t slots, under the condition the arrival process changes from state r

|0( )

c

r r

m

0 (=H or L) prior to the beginning of the interval, to state r_t (=H or

, (6)

, is the probability that the arrival process changes from state x to state r_t . The first term within the square bracket in Equation (6) corresponds to that all m packets arrive in the first t-1 slots and no packet arrives in the last slot.

The second term represents that m-n packets arrived in the first t-1 slots and a batch of

n ⁿ

^≤

m packets that arrive in the last slot with probability )

λr_t⋅b nr_t^{( )}.

With the “

( )

” sign removed, Equation (5) can be expanded into three cases, as

Notice that is absent from the first case of Equation (7) due to that the inter-departure time is zero if a departing burst leaves behind a system with

ψ

or more packets. we now compute the queue length distribution by first conditioning on the value of and separating case one from cases two and three in Equation (7), as

(9)

To proceed,

1| |

k k k

z y y

P u⎡⎣ ₊ =u q =q⎦ in Equation (10) needs to solve first. It can be resolved by considering five cases depending on different ranges of u and q values as given in Equation (11) below. First of all, in case (1) when q≥ a

ψ

full-size burst is immediately transmitted, yielding

t

=0. Thus, the probability under

u = 0 is one. In case (2), when

0< <q ψ but u+ ≥ , the total number of packets

q ψ

in the queue must exceed

ψ

the first time at a particular slot before the BATr expires.

Namely, within an interval t of less than or equal to

τ

, there arrives a total of m packets during t-1 slots, and exactly at this final slot, a batch of

(

⁰≤ ≤ − −

^m ψ ^q

)

That is, the total number of packets that arrive within an interval of

τ

is u busy period of the system. Notice that BATr is not activated until the arrival of the first batch with m packets. This explains the term within the square bracket. Under such condition, this case becomes identical to that when a departing burst leaves behind a system with m packets, with the probability shown before the product sign. Notice that, this probability can be obtained by applying cases (1) to (3) once, depending on the m value. Combining the results from the cases discussed above, it has

With Equations (6) and (8)-(11), the limiting queue length distribution under the arrival process being at state H or L, can be given by

{ }

lim , ,

y y

P q d

P q d y H L

→∞ ⎡ ⎤

⎡ = ⎤= = ∈

⎣ ⎦ ⎣^k ⎦ . (12)

We are now in the position to determine the departure process distribution,

( )

P

t s

t s

. There are four cases depending on different t and s values to be considered.

First, in Case I when , it is clear that the queue length is larger than

ψ

behind the burst departure. We get that

t

Case I: t

( )

^{ ^,^}

, ,

0 , if

y H L

P

t s

t s

s

, if

P q

ψ s ψ

ψ

= ⎨ ∈

⎪

⎧ ⎡⎣ ≥ ⎤⎦ =

⎪

⎩ <

∑

t

(13)

Second, Case II corresponds to the transmission of a full-size burst due to having a total of

ψ

or more packets before the BATr expires. Hence, we obtain that

τ

< <

( ) ( )

1 , if

c

m P b n P q q s

Case II:

( )

, , _{ ,^;_} ^| ^,

, , .

0 , if

i y i j j j y

q m n q m

y i j H L

P

t s

t s

s

ψ ψ

λ ψ

ψ

⎧ ⎡ − ⎤

⎡ ⎤

⋅ ⋅ ⋅ ⋅ = =

∑ ∑

t

+ < ≥ − −

∈

⎪ ⎢ ⎥ ⎣ ⎦

⎪ ⎣ ⎦

= ⎨⎪

⎪ <

⎩

(14)

Third, in Case III when = and

τ

s=ψ , the total number of packets in the system exceeds

ψ

exactly at the same time when the BATr expires. Otherwise, if

s<ψ , a burst of size less than

ψ

is transmitted due to BATr time-out. That means,

( )

^{ ^}

Finally, under the last case when > , the departing burst must have left

τ

an empty system

( ^{P q}

^⎡⎣^y ⁼⁰^⎤⎦

)

t

resulting in the deactivation of the BATr. The timer remains deactivated until the arrival of the first batch of packets. Then, whether the next departing burst is a full-size one or not depends on the total number of arriving packets, as

Case IV:

τ

Combining Equations (13)-(16), we achieve the joint-form departure process distribution.

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ψ

τ

ψ

τ

α

β

λ

λ

α

β

λ

λ

i j

{ H L

}

b

b

b b

B L b b

λ λ

β λ α λ

α β α β

t

Legend:

: Imbedded Markov chain epochs;

0

System setting:

ψ = 4;

BoB : Begin-of-Burst;

EoB : End-of-Burst;

t

s

q

v

τ

t

τ

τ

ψ

q

ψ

q

q

v

( ) {

}

q q u v

ψ

{

}

y y

z

H L ( )

{ }

ψ

{

{

}

}

c

m

n n

m packets that arrive in the last slot with probability )

( )

ψ

ψ

t

u = 0 is one. In case (2), when

q ψ

ψ

τ

(

m ψ q

)

τ

{ }

P q d

P q d y H L

( )

^{i j}

{ ^{H L}

( ) ^{

^}

n ⁿ

^m ψ ^q

( ^{P q}