A Pseudo-Bayesian Aloha Algorithm with Mixed Priorities for Wireless ATM Jean-Franc¸ois Frigon and Victor C.M. Leung Department of Electrical and Computer Engineering, University of British Columbia E-Mail: fjeanf,
[email protected] ABSTRACT Uplink
Downlink
In reservation MAC protocols, before obtaining a contention free access to the channel, a mobile must wait for its request packet to be successfully sent to the base station. A pseudo-Bayesian Aloha algorithm with multiple priorities is proposed in this paper to reduce the waiting time of delay sensitive request packets in a multimedia environment. Packets are transmitted in each slot according to a transmission probability based on the channel history and a priority parameter assigned to their priority class. An adaptation of the slotted protocol to the framed environment is also described. Simulation results are presented and show that the protocol offers a significant delay improvement for high priority packet with both Poisson and self-similar traffic while low priority packets only experience slight performance degradation. I.
INTRODUCTION
In reservation MAC protocols, a mobile sends a request to the base station to obtain a contention-free access to the wireless channel. However, in most of the systems, the request packet is sent in contention with request packets from other connections. Thus, the most critical part for quality of service and delay in a reservation MAC protocol is the contention phase that a connection go through before obtaining contention-free packet transmissions. For certain classes of traffic the contention phase is the limiting part of the scheme while other classes are less sensitive to the introduced delay. The first packet of a voice talkspurt, a request for new bit-rate in a VBR connection or a handoff request for a real-time connection are examples of control packets sensitive to contention delay. On the other hand, control packets for new data messages or request to establish a new connection are less time sensitive. It is thus necessary to find a contention access protocol that will give priority to delay sensitive packets in order to implement an efficient multimedia wireless MAC protocol like wireless ATM. Several contention protocols with mixed priorities are presented in the literature. Some protocols avoid collisions between high and low priority packets [1, 2, 3]. They postponed low priority packets transmissions until they detect, through channel feedback, that there is no high priority packet in the system. Other protocols allow collisions between high and low priority packets [4, 5, 6, 7] and service priority is given by the collision resolution protocol. This work was supported by a postgraduate scholarship from the National Sciences and Engineering Research Council of Canada.
1 2
..
N
Downlink Control Slots
...
Data slots
1 2
..
Uplink Control Slots
N
...
Data slots
Figure 1: Multimedia wireless MAC protocol frame structure
However, the proposed priority protocols work on a slot basis and required an immediate feedback after the contention slot. But, this is not possible in reservation protocols where a frame structure similar to the one illustrated by Figure 1 is used. A straightforward modification that can be implemented is to form N different sessions for each uplink control slot. Then, feedback information sent in the downlink control slots can be used to update each independent session. Even if this modification will maintain the maximum global throughput, this separation between sessions is not desirable since it will cause longer delays. There is thus a need to find a new random access protocol with mixed priorities that will be specifically adapted to the reservation protocols structure. For example, these protocols are centrally controlled, hence information about the actual state of the multiple access algorithm can be maintained allowing an easy implementation of a full-sensing algorithm. Furthermore, any deterministic algorithm can not be used since we can not make any assumption about the user population size. Finally, we must remember that in reservation protocols the overall throughput is not determined by the random access throughput, but the quality of service is highly dependent on the delay encountered by control packets. Hence, we should put more emphasis on the access delay than on the throughput. We thus want to have a relatively constant low delay for high priority traffic in a relatively wide range of total traffic rates , without introducing an excessive delay for low-priority traffic. To implement this new protocol, we have chosen to explore the possibilities offered by the slotted Aloha protocol. In the next section, we will present a slotted Aloha protocol with mixed priorities. A modification to adapt the protocol to the framing environment will then be introduced in Section III. Finally, simulation results are presented in Section IV.
II.
PSEUDO-BAYESIAN SLOTTED ALOHA WITH PRIORITIES
each priority class i, including new arrivals, after either an idle slot or a success slot, are also independent and Poisson distributed with parameter nˆ i γi + λi .
It is known that the basic slotted Aloha algorithm, where a node transmits new packets when it receives them and retransmits backlogged packets with a fixed probability qr , is unstable for any value of the arrival rate. Thus to implement a slotted Aloha with priority classes, we have derived an algorithm similar to the pseudo-Bayesian Aloha stabilization algorithm presented in [8] and [9]. Let new packets be regarded as backlogged immediately after their arrival. They will attempt transmission in subsequent slots until success with a probability determined by their priority class and the estimated backlogged state of the system. The cumulative input arrival process consists of p independent Poisson processes with intensities λ1 ; : : : ; λ p . Let n1 ; : : : ; n p and q1 ; : : : ; q p respectively be the number of backlogged packets and the transmission probability of each traffic classes. Then, the channel traffic generated by class i is Gi (ni ) = ni qi and the total attempt rate is p G(n1 ; : : : ; n p ) = ∑i=1 ni qi . The probability that a packet of the ith traffic class is successfully transmitted in a slot is then given by: i Psucc Gi (ni) e
G(n1 ;::: ;n p )
(1)
and the probability that a packet from any class is successfully transmitted is: Psucc G(n1 ; : : : ; n p ) e
G(n1 ;::: ;n p )
Now, assume that before a slot the numbers of backlogged packets of each priority class i are statistically independent and are given by a Poisson distribution with parameter nˆ i γi . Furthermore, each packet of class i (1 i p) is independently transmitted in the next slot with probability qi = γi =nˆ i . 0
Let ni denote the number of backlogged packets of class i (1 i p) after a slot (excluding new arrivals). Therefore, we 0 can show that the ni ’s joint and marginal probability distributions, given that the slot was either idle or a packet from priority class j (1 j p) was successfully transmitted, are: p
0
p(n1 ; : : : ; n p j idle or succ j ) = e ∏ i=1
0
p(ni j idle or succ j ) =
(nˆ i
(nˆ i 0
γi )ni e 0 ni !
0
0
p(n1 ; : : : ; n p j coll) = 0
p
∑
γi )ni
e e
0
γi )ni e 0 ni ! (nˆ i γi )
nˆ i
(3) (4)
Furthermore, the arrival process is Poisson and independent of the contention system. Thus, the numbers of backlogged packets of
p
e nˆi ∏ 2 i=1 n0i ! 0
0
ni γi (nˆ i
γi )ni
0
1
p(ni j coll) =
e 0
+ ni γi (nˆ i
γi )
eγi 0
ni 1
1
0
p
∏ nˆi i i=1
∏(nˆ j j=1 j 6=i
e nˆi n0i nˆ 2 n0i ! i
e
p
p
i=1
h
(2
n
∏(nˆi i=1
0
(5)
γ j )n j
γi)(nˆ i
0
γi )ni
i
(6)
We can clearly see that after a collision slot, the numbers of backlogged packets in each priority class i are not independent and Poisson distributed. However, we can find that the distribution of the number of backlogged packets for each priority class i after a collision slot, including new arrivals, is reasonably approximated as a Poisson distribution with parameter nˆ i + γi =(e 2) + λi. We can also find that the correlation between the number of backlogged packets of two different priority classes i and j (i 6= j) is given by:
(2)
We see that if G(n1 ; : : : ; n p ) is maintained at the optimal value of 1, the system can achieve its maximum throughput of 1=e. The priority class i throughput will then be Gi (ni )=e. There is thus a possibility to adjust the throughput of each stream to a specific value. Let γ1 ; : : : ; γ p be the priority parameter of each traffic class p and Gi (ni ) = ni qi = γi . If we impose the constraint ∑i=1 γi = 1 we obtain the desired maximum throughput 1=e and each traffic class has a throughput γi =e.
0
0
For the case where a collision occurred in the slot, the ni ’s joint and marginal probability distributions are:
ρ= (e
2)2
γi γ j
nˆ i + e
γi
γi 2
e 2
2
nˆ j +
γj e 2
γ j 2 e 2
1=2
(7)
Since γ 1 and nˆ is likely to be large when there is a collision, we observe that the correlation will be negligible. Furthermore, the arrival streams for each traffic class are independent. We can 0 thus reasonably assume that the ni ’s are independent. Therefore, our initial assumptions on the independent and Poisson distributions of the number of backlogged packet of each traffic class i (1 i p) are satisfied for the three possible slot outcomes: idle, success or collision. Algorithm Based on the obtained results we can derive an algorithm to implement a multiple access protocol with mixed priorities. Suppose that we have p different priority classes with independent Poisson arrival processes of intensities λ1 ; : : : ; λ p . A lower index corresponds to a higher priority packet class. Let γi be the priority parameter of each traffic class i. In order to maintain the priority order we must have γ1 γ2 γ p 1 γ p and p the parameters satisfy the relation ∑i=1 γi = 1. The algorithm operates by maintaining for each priority class i an estimate nˆti of the number of backlogged packets nti at the beginning of each slot t. For each priority class i, an effective priority parameter γˆti is also computed (used to avoid that γi nˆti ). A new arrival during slot t is immediately regarded as backlogged and it will attempt transmission in each subsequent slot after its arrival until success. At the beginning of each slot t, nˆti is updated from nˆti
1
, γˆti
1
and
the feedback for slot t (
nˆti =
max(λi ; nˆti
1 according to the rule: 1
+ λi
γˆti 1
γˆti
1
nˆti 1 + λi + e 2
) for idle or success for collision
(8)
When the estimated number of backlogged packets for each traffic class has been computed, we find the effective priority parameter of each class. The proration algorithm ensures that each class is assigned its priority parameter if γi nˆti , or its estimated number of backlogged packets, otherwise. Then, the algorithm distributes the “leftover”, in order of increasing priority, to each class up to a maximum equal to their estimated number of backlogged packets. If there is still a “leftover” it will be assign to each class proportionally to their arrival rate. The process can be summarizes as follows:
The arrival rate λi for each priority class i (1 i p) is given in number of packets per frame. The same definitions that were presented in Section II for γ, γˆ and priority order are assumed. The algorithm operates by maintaining for each priority class i an estimate nˆti of the total number of backlogged packets nti at the beginning of each frame t. A new arrival during frame t is regarded as backlogged and it will attempt transmission in each subsequent frame after its arrival until success. At the beginning of each frame t, for each priority class i, nˆti is updated from nˆti 1 , γˆti 1 and the feedback (let nnc be the number of idle or success slots and nc the number of collision slots in frame t 1) for frame t 1 according to the rule:
nˆti = λi + nnc max 0;
nˆti 1 K
γˆti
1
+ nc
nˆt 1 i
K
+
γˆti 1 e 2
(10)
for (each priority class i in order of increasing priority)
γˆti = min nˆti ; 1
p
i 1
∑ γˆti j=1
∑
When the estimated number of backlogged packets for each traffic class has been computed, we find the effective priority parameter of each class using the proration algorithm presented in the previous section where nˆti is replaced by nˆti =K. Then each backlogged packet is independently transmitted in frame t according to the transmission probability qti of the priority class i it belongs to. Transmission probabilities are calculated as follows:
min(nˆtj ; γ j )
j=i+1
p
L=1
∑ γˆti i=1
for (each priority class i) λi γˆti = γˆti + p L ∑ j=1 λ j
Then each backlogged packet is independently transmitted in slot t according to the transmission probability qti of the priority class i it belongs to. Transmission probabilities are calculated as follows:
γˆt qti = min 1; it nˆ i III.
(9)
FRAMED ALOHA WITH PRIORITIES
To adapt the slotted algorithm to the frame structure, we propose a strategy where an arrival packet waits until the end of the ongoing frame before attempting its first transmission (gated system). Starting from the next frame, it will independently attempt to transmit with a probability qr in a randomly chosen slot for each frame. The selected slot is independent from frame to frame. Let suppose that there is K slots in a frame and the a priori distribution of the total number of backlogged packets n j at the beginning of frame j is Poisson with parameter nˆ j . If each backlogged packets independently chooses a given slot in the frame for transmission, then the distribution of the number of backlogged packets nij that chose a given slot i is Poisson with parameter nˆ j =K and independent for each slot i. Then using, the results from the previous section, nij+1 , the numbers of backlogged packets in each slot i, are independent Poisson random variables with parameter nˆ ij+1 . Hence, n j+1 , the total number of backlogged packets at the beginning of frame j + 1, is also Poisson with mean ∑Ki=0 nˆ ij+1 which satisfies our initial assumption. Algorithm Using these results, we can derive from the pseudoBayesian priority algorithm presented in the previous section a multiple access protocol with mixed priorities for a K slots frame.
qti
γˆt = min 1; it K nˆ i
(11)
If a packet is transmitted in a given frame, it will independently choose a given slot for transmission with an uniform probability (each slot has a probability 1=K of being chosen). IV.
SIMULATION RESULTS
In this section we present the simulation results for the slotted and framed pseudo-Bayesian priority algorithms. To evaluate the proposed systems, the delay measurements have been compared with results obtained when only one traffic class is considered (“Basic PB”). The arrival rate value used by the backlogged estimation algorithm is an estimate computed with the moving time-average of successful transmissions for each traffic class over a window period of 500 slots. We have simulated each experimental condition for a period of at least 10 million slots. The framed scheme has been simulated with ten slots per frame and delays calculated in number of frames are presented for this scheme (waiting time in slots has no signification in the framed environment). Figure 2 shows, for the framed system, the effect of the priority parameter on the waiting time for fixed arrival rates. As we expected from the throughput equations presented in Section II, we observe that a class has a delay advantage when its arrival rate is smaller than its share of the total throughput (i.e. λi γi = ∑ pj=1 γ j ). An interesting phenomenon can also be observed: the delay experienced by both traffic class decreases as we move away from the fair share point. Thus, the optimum operating point is at γ1 = 1. Similar results are obtained with the slotted system and for other values of arrival rate and a greater number of traffic class.
10
8
Class one Class two Average Basic PB
7
Class one Class two Average Basic PB
9
Waiting time in number of frames
Waiting time in number of frames
8
6
5
4
7
6
5
4
3
3 2
2
1
0
0.1
0.2
0.3
0.4 0.5 0.6 0.7 Traffic class one priority parameter
0.8
0.9
1
Figure 2: Waiting time versus the priority parameter for the framed system. λ1 = 0:15 and λ2 = 0:20 packets/slot.
0
0.01
0.02
0.03
0.04 0.05 0.06 0.07 0.08 Traffic class one arrival rate (packets/slot)
0.09
0.1
Figure 4: Waiting time versus traffic class one arrival rate for the framed system. λ2 = 0:25 packets/slot and γ1 = 1. 1
30
Class one Class two Average Basic PB
25
0.9
0.8
Class one Class two Average Basic PB
0.6 P( y
