Chapter 5 Optimal Queue Management Algorithm for Real-Time Traffic
5.2. The Proposed PL Queue Management Algorithm
It is assumed that packets are infinitesimally dividable, which is referred to as fluid-flow model in this dissertation. A more realistic system which manages the queues packet by packet, namely, packet-based system, will be studied in the next section. Consider f , k 1 k K, in the n time th
For better comprehension, we firstly investigate the case that all traffic flows request identical delay bound and then extend the results to a more general case that traffic flows request different delay bounds, which completes the description of the proposed PL queue management algorithm.
Flows with identical delay bound requirement
Assume that k , 1 k K, where is a positive integer. Consider the n time slot. th
Define the running loss probability of f as k P nk
L nk
A nk . Upon traffic arrivals, all data buffered in the multiplexer is schedulable iff
where U denotes the set containing traffic flows with non-empty queues. If equation (41) holds,
we have l nk
0 and thus L nk
can be updated by L nk
L nk
1
, 1 k K. Assume space-conserving if equation (42) holds for each time slot, assuming that all traffic flows request identical delay bound.To satisfy both equations (43) and (44), we found that one simple interpretation, called water-filling interpretation, can be adopted to facilitate the development of the proposed PL queue
management algorithm. As an example, assume that U { , , , , }f f1 2 f3 f4 f5 , , all flows 1
belonging to U have packets arrived in the n time slot and th Loss n
0. Before performing queue management, we interpret the state of each flow by the picture portrayed in Fig. 5.2 (a). In this figure, notice the following observations. 1) For each flow belonging to U, there is a rectangular vessel, which consists of two parts, solid (gray) and hollow (white) part. Note that the bottom lengths of all rectangular vessels in the x axis are assumed to be the same and thus it sufficesto consider a 2-D picture. 2) Consider f . The volumes of its solid and hollow part are equal to 1
1 1
L n and Q n1
, respectively, while the common bottom areas of them are the same and equal to P A n1 1
. In other words, if we fill some water with amount x, 0 x Q n 1
, into this vessel, the level of water will be increased up to
L n1
1
x
A n P1
1
. The same idea can also be applied to all the other flows belonging to U. Based on these two observations, we found that managing queues so that both equations (43) and (44) are satisfied is just identical to fill water withamount Loss n
into the super vessel, the combination of all the vessels of flows belonging to U.The results are shown in Fig. 5.2 (b). In this figure, it is not hard to see that the level and volume of water contained in the vessel of each traffic flow represent, respectively, the corresponding running loss probability and loss amount in the n time slot. th
1 1
P A n P A n2 2
P A n3 3
P A n4 4
1
Q n
2
Q n
Q n3
4
Q n
5 5
P A n
Q n5
Fig. 5.2(a) Initial state
1 1
P A n P A n2 2
P A n3 3
P A n4 4
P A n5 5
l n1 l n2
l n3
5
l n
Fig. 5.2(b) The result of water-filling
Fig. 5.2 An example for illustrating PL queue management algorithm for traffic flows with identical delay bound requirement.
Inspired by the water-filling interpretation, we develop the proposed PL queue management algorithm by placing all flows belonging to U into their appropriate subsets (U , C U , or P U ) and Z
then calculating their individual loss amount. Define
phase of the proposed PL queue management algorithm decides which flows should be placed inU . Define C H , k 1 k K , as
Note that H represents the capacity if the super vessel is filled with water (lost data), where the k level is up to F . Therefore, we have k fkUC iff Hk Loss n
. Since, by assumption,The second phase of the proposed PL queue management algorithm decides which flow should be placed in U . It is not hard to see that P UP and e e UP. As a result, the remaining
Note that H represents the capacity if the super vessel is filled with water (lost data) up to the ej level S . We have j fjUP iff 1) fj or 2) e fj and ( e e) Hej Loss n
. AfterU and C U are determined, one can obtain P UZ U UCUP.
Once all flows are placed in U , C U and P U appropriately, we have Z
Note that the derivations of equation (51) is basically the same as those of equation (22) and thus are not repeated. Finally, discard l nk
from the head of Queue and update k L nk
by
1
k k k
L n L n l n for each fk , which completes the proposed PL queue management U algorithm. To facilitate the presentation in the next sub-section, the above procedure based on the
water-filling interpretation is represented as
Flows with Different Delay Bound Requirements
Without loss of generality, let U { , ,...,f f1 2 fK} and 12 ....K. Assume that all data buffered in the multiplexer is schedulable in the (n1)th time slot and some data of f arrives in k the n time slot. In the th n time slot, it is not hard to see that all data buffered in the multiplexer th
is schedulable iff delay bound has higher priorities than the newly-arrived one of f . Similarly, if multiple flows k have data newly arrived, all data buffered in the multiplexer is schedulable iff equation (53) holds for
min, min 1,..., K
j , where min min { }
k new
f U k
and Unew is a set which contains traffic flows
with newly-arrived data. Again, if all data in the multiplexer can be transmitted before their own
deadlines, we have l nk
0 and thus L nk
can be updated by L nk
L nk
1
, 1 k K. can be buffered in the multiplexer without violating their delay bound for no longer than i time slots.Denote l nki
, 1 k K, as the corresponding loss amount of data belonging to f , and we have kAssume that Loss ni
0. Obviously, l nki
0 if
im1Qkm
n
im10lkm
n 0. Definecorresponding inputs and outputs of which are described as follows
1
11
1
1 1
11
1 1
Fig. 5.3 illustrates an example considering flows with different delay bound requirements.
Assume that there are five traffic flows in the multiplexer, where
1, 2, ,3 4, 5
1,1, 2,3,3
. Each flow has data arrived in the nth time slot and it holds that Q n31
0, and Qkm for 0
k m,
4,1 , 4, 2 , 5,1 , 5, 2 . After plugging the related information into equation (54), wehave Loss ni
0 for i1,3 and Loss n2
0, meaning that WF needs to be performed for two rounds. Fig. 5.3(a) shows the initial state of each flow in the first round. The results of the first round are presented in Fig. 5.3 (b), which in turn to be the initial state of each flow in the second round. In the first round, it holds that U { , , , }f f1 2 f4 f5 , UC { , }f4 f5 , UP { }f1 and{ }2
UZ f . Similarly, Fig. 5.3 (c) demonstrates the results of the second round. In the second round, we have U { , , , , }f f1 2 f3 f4 f5 , UC , UP { , }f3 f4 and UZ { , , }f f1 2 f5 .
It is clear that the theoretical results developed in [11] can also be applied to prove that the proposed PL queue management algorithm, coupling with EDF service scheduler, is optimal in the sense that the effective bandwidth is minimized under generalized space-conserving constraint. In fact, we have shown the proposed PL queue management algorithm is generalized space-conserving and the criterion shown in equation (43) is identical to that of G-QoS scheme, assuming the size of cell is infinitesimally small.
3 3
P A n P A n4 4
P A n1 1
P A n2 2
P A n5 5
1
l n4
1
l n1
1
l n5
Fig. 5.3 (b) The result of the first water-filling
3 3
P A n P A n4 4
P A n1 1
P A n2 2
P A n5 5
1
l n4
1
l n1
1
l n5
3
l n3 l n43
Fig. 5.3 (c) The result of the second water filling
Fig. 5.3 An example for illustrating PL queue management algorithm for traffic flows with different delay bound requirement.