Analysis

In this section, we present an analytical formula to calculate the average gap cessing time of the indicator-based stall avoidance mechanism. The average gap pro-cessing time is an important performance metric for the high speed retransmission mechanism in both the MAC and RLC layers. For example, if a Type-II gap occurs, the received packets are queued in the reordering buffer of the MAC layer. Thus, a longer gap processing time causes a higher overflow probability in the MAC layer

reordering buffer. Furthermore, a Type-II gap will trigger an RLC retransmission.

Thus, if the gap processing time is too long, a large-sized buffer in the RLC layer is required to accommodate the packets forwarded from the MAC layer. However, it is difficult to evaluate the gap processing time of the indicator-based stall avoidance mechanism in an analytic way. The gap processing time is a function with parameters from both the physical and the MAC layers. For example, in the physical layer, the packet error rate and the probability of a NACK becoming an ACK should be incor-porated in this function, while in the MAC layer, the impact of the scheduling policy on the gap processing time should also be considered. Hence, to make the analysis tractable, we has made the following assumptions:

1. A fair scheduler independently assigns each process to each user with a proba-bility of Psch= 1/K, where K is the number of users in the system.

2. Because a NACK-to-ACK error usually occurs when a mobile terminal moves at high speeds, it is assumed that the fast changing channel is modelled by an independent Rayleigh fading channel from one packet to another packet.

3. In the receiving end, a reordering buffer is assigned to a user to handle the received packets from multiple parallel HARQ processes.

4. Effects of incremental redundancy and Chase combining have not been consid-ered in the analytical model yet. Thus the provided analysis can be viewed as a worst-case analysis compared to the cases applying incremental redundancy and Chase combining.

5. The feedback delay of sending an ACK or a NACK in an individual HARQ process is not taken into account of the gap processing time. In HSDPA, mul-tiple parallel HARQ processes transmit data packets alternately to fully utilize the channel capacity. Thus the feedback delay of sending control signals does

not affect the gap processing time. In the feedback channel, only the impact of NACK-to-ACK errors is considered.

Now we prove that the average gap processing time of the indicator-based stall avoidance mechanism for the multiple parallel HARQ processes can be calculated by the following proposition.

Proposition 1: Consider an V -process SAW HARQ process. For a given packet error rate (P_e) and the probability of an NACK-to-ACK error (P_{N →A}), define the probability of generating new packets and old packets as

P_new = (1 − P_e) + P_eP_{N →A} (4.9) and

P_old = P_e(1 − P_{N →A}) , (4.10)

respectively. Then, the gap processing time for the indicator-based stall avoidance method can be calculated as follows:

GP T =

The parameter C_V in (4.11) is the required cycles to involve all processes in the V-process SAW HARQ to remove a Type-II gap, and the other parameters V , Psch, and P (x_n= S_T) are already defined in Section 4.1.

Proof: Assume that a Type-II gap appears in cycle 0 of process P R₁ as shown in Fig. 4.4. We now consider the following two possible scenarios to calculate the average gap processing time.

(I) Type-II gap can be removed at process 1:

When all the SAW HARQ processes except P R₁ enter the STOP state (S_T), the gap will be removed at process 1 when process 1 enters the STOP state in the future. Process 1 will transmit a new packet whenever it is scheduled to transmit a packet in the k-th cycle because the NACK signal for the missing packet is contam-inated to be an ACK signal. Hence, this new packet sent by process 1 is associated with a NEW NDI state. Consequently, the state of process 1 enters the STOP state at k-th cycle. Since all (V ) processes enter the STOP state and the gap of the missing packet is still in the reordering buffer, the receiver can judge that the missing packet is a Type-II gap and will never be retransmitted. In this case, the receiver takes a period of kV TTIs to remove this Type-II gap from the reordering buffer, where other (V − 1) processes turn to the STOP state before k-th cycle. Denote E_A and E_B the event that process 1 being scheduled for transmission at the k-th cycle and that all other (V − 1) processes enter the STOP state before the k-th cycle, respectively.

Then, it is followed that

P (EA) = Psch(1 − Psch)^k−1 (4.13) and

P (E_B) =

"_k−1 X

i=0

P (x_i = S_T)

#_{V −1}

. (4.14)

Recall that

P (xn = ST) = P (xn−1= S1)Psch+ P (xn−1= S2)Psch(1 − Pe) , n ≥ 1 . By iteratively substituting (4.2)-(4.6) into (4.7), we obtain

P (x_n= S_T) = P_oldP_sch[P_eP_{N →A}+ P_old(1 − P_e)] [P_schP_old+ (1 − P_sch)]ⁿ⁻¹, n ≥ 1 . (4.15)

Because events EA and EB are mutually independent, the probability of the Type-II gap being removed in the k-th cycle of process 1 can be expressed as

P (E_A∩ E_B) = P (E_A)P (E_B) . (4.16) Combining (4.13), (4.14), and (4.15), the average gap processing time to remove a Type-II gap at process 1 is

GP TI =

CV

X

k=1

kV × P (EA)P (EB)

=

CV

X

k=1

kV × Psch(1 − Psch)^k−1

"_k−1 X

i=0

P (xi = ST)

#_{V −1}

, (4.17)

where C_V is the required cycles of involving all processes in the V-process SAW HARQ to remove a Type-II gap.

(II) Type-II gap can be removed at process m, for m ≥ 2:

Assume that the Type-II gap is removed in the k-th cycle of process m, where m ≥ 2 and k ≥ 1. In this case, the gap processing time is (m−1+kV ) TTIs, as shown in Fig. 4.4. Denote P_α(m, k) as the probability of process m removing the Type-II gap in the k-th cycle for different combinations of k and m. Then, the average gap processing time for the Type-II gap being removed at process m, where m ≥ 2, can

be expressed as where CV is defined in (4.17). The probability Pα(m, k) is the joint probability of following four independent events: Similar to the procedures of deriving (4.13) and (4.14), we can rewrite (4.19) as



Substituting (4.20) and (4.21) into (4.18), we can have the probability of removing

T he f irst m−2 processes

z }| {

T he last V −m processes

z }| {

Note that the value of CV can be obtained by satisfying the following equation:

CV With (4.17) and (4.22), the average gap processing time for the indicator-based stall avoidance mechanism with multiuser communications is equal to

GP T = GP T_I+ GP T_II

Fig. 4.4: An illustration for the gap processing time of the indicator-based stall avoidance mecha-nism.

¥ The accuracy of the analytical formula for estimating the gap processing time will be validated by simulations in Section 4.3.