1. Introduction
Direct-sequence code-division multiple access (DS CDMA) is a wireless transmission technology which offers random access abilities and potential for higher throughput [1-7]. In [8-15], the DS CDMA/slotted ALOHA system is investigated in wireless and optical channels. In [16-20], the throughput of the DS CDMA/unslotted ALOHA system is analyzed in AWGN channels. The throughput analysis of the DS CDMA/unslotted ALOHA wireless network in Rayleigh fading channels has been conducted in [21-23]. They all assumed packet-level analysis, and ignore that the channel gain and/or multiple access interference level changes symbol-by-symbol within the packet. In [24], we proposed the finite-state Markov channel (FSMC) to analyze throughput performance of the uncoded DS CDMA unslotted ALOHA networks with fixed length data traffic. The channel gain as well as multiple access interference can vary from symbol by symbol.
In this paper, we propose the FSMC to analyze throughput performance of the coded DS CDMA unslotted ALOHA systems for variable message length with one or two user classes, and access control in mobile channels. This is an extension of our previous work in [24]. We add channel coding, variable length data traffic, and two user classes in this paper. We assign different priorities to two user classes by different packet transmission probabilities (access control), which improve the throughput of the higher priority user class in high offered load [18][19].
The significance of this extension is as follows:
The rest of this paper is organized as follows: In Section 2, the FSMC model is briefly introduced. In Section 3, the system model is presented. In Section 4, we analyze the throughput of the DS CDMA/unslotted ALOHA networks in Rayleigh fading FSMC channel for variable length data traffic and one/two user classes. The simulation and numerical results are given in Section 5. Section 6 is the conclusion.
2. FSMC Model (1-D Markov process)
In many cases, modeling a Rayleigh fading channel as a two-state Gilbert–Elliot channel [25][26] is not sufficient because the channel condition can vary dramatically. Extending the two-state model to a finite-state one is called FSMC model [27][28]. FSMC model depend on the condition that the users’ received signal-to-noise ratio (SNR) may be highly time-varying in mobile channels.
Because a traditional time-varying Rayleigh fading channel, produce time-varying received SNR, by dividing the range of the received SNR into a finite number of interval, a FSMC model can be built for the Rayleigh fading channel. A finite state Markov chain channel is defined by its transition probabilities. We denote discrete-time Markov chain channel with state space {1,2,…,M} and transition probability matrix h = [hi,j], 1 ≤ i,j ≤ M . The hi,j is defined as the probability that the Rayleigh fading channel changes from the state i to the state j. Let A denote the received SNR that is proportional to the square of the signal envelope. The probability density function (PDF) of A is exponential and can be written as
Where ξ is the expected value of A .Let 0 = A0 < A1 < ⋯ < AM-1 < AM = ∞ be the thresholds of the received SNR. Therefore, the Rayleigh fading is said to be in the state j , j = 1, 2, …, M , if the received SNR is in the interval [Aj-1, Aj). Nj , j ∈ {1,2,⋯, M} is the expected number of the received SNR level crossing downward the boundary Aj at state j of FSMC, and can be written as
fm is the maximum Doppler frequency and given by
where v is the vehicle speed and λ is the wavelength.
We assume the interference levels depend on consecutive symbols are neighboring states. Then, the element of the transition probability matrix H is given by [27]
We used a BPSK modulated communication system with a transmission rate of R bits per second. The average bit per second transmitted during which the radio channel is in state j is
where the steady state probability for each state is
Finally, we can get the approximated value of the transition probability as follows [27]
3. System Model
We consider a single-carrier DS CDMA system with random spreading codes of length N in an unslotted ALOHA network for two user classes with variable message length. Each class’s message has Bn (n = 1,2) fixed-length packets, where Bn is geometrically distributed with expected value are the average number of packets in a message for class 1 and class 2, respectively .The length of packet is fixed to be L bits, so the message length of class 1 and 2 are B1L[bits] and B2L[bits], respectively. We consider a single hub station and an infinite number of independent users in our system network. We use a Poisson process with an arrival rate λ1 for class 1 and arrival rate λ2 for class 2 to model the arrival process of the system for our system network.
The offered traffic load for the class n, of the network is defined as Gn (packets/packet duration), n=1, 2. The corresponding throughput is defined as Sn (packets/packet duration) , n=1, 2. Finally, the system throughput is defined as Stot = S1 + S2, respectively. It’s easy to obtain the packet duration Tp[sec] by L multiplied by 1/R , where R [bits/sec] is the data rate of system.
The bit-error probability of an asynchronous DS CDMA system [29] for k interfering users and FSMC channel state j by simplified improved Gaussian approximation is expressed as
where
where N0/2 is the two-sided power spectral density of Gaussian noise, Eb is the bit energy and Q(x) is given by
The third term of (7) may not be real-valued, if k is too small. If this occurs, we can use standard Gaussian approximation [30] instead,
In (7) and (9), bit errors in a packet are caused by the effect of the multiple access interference (MAI) and Rayleigh fading. The hub station decides the packet transmission probability Ptr,n of class n users based on the offered load of class n users, Gn , n=1, 2, and then broadcasts the packet transmission probability to mobile stations (MSs). The total offered load must be less than Gmax, where Gmax is the maximum offered load the system can reach. The packet transmission probabilities for low, medium, and heavy load are as follows [18][19]
4. Throughput Analysis For DS CDMA/UNLOTTED ALOHA System Combining FSMC Model
4.1 M/M/∞ Queueing Model
The DS CDMA/unslotted ALOHA system with variable message length and one/two user classes can be modeled as the M/M/∞ queuing model, as stated in our previous works in AWGN channels [19][20]. Here we briefly review the reasoning. We first recall a theorem from [31].
Theorem1[31]:For M /G /∞ queueing model with arrival rate λ, the probability distribution for the number of packet in the system is given by
where 1/μ is the average packet length.
The above theorem from [31] states that the number of users in M/G/∞ queuing system is function of the average packet length only, not the distribution of packet length. Thus M/M/∞ (exponential packet length) is equivalent to M/D/∞ (fixed packet length) for DS/CDMA unslotted ALOHA systems because the packet success probability depends on the number of users only (the equivalence may not be true for other communication systems). Thus, instead of the M/D/∞ queueing model with the death rate approximation in [17] (variable message length and one user class) [18] (variable message length and two user classes), we propose M/M/∞ queueing model for the DS CDMA unslotted ALOHA system [20] (variable message length and one user class) [19] (variable message length and two user classes). As shown in [17-20], when one message is equal to variable number of fixed length packets (variable message length), the analysis of variable message length in AWGN channels follows (or bases on) the analysis of fixed packet/message length (one message=one packet) in AWGN channels [16]. Therefore the DS CDMA/unslotted ALOHA system with variable message length and one/two user classes in FSMC fading channels can be modeled as the M/M/∞ queuing model like our previous works in AWGN channels [19-20].
4.2 Proposed Throughput Analysis for Variable Message Length and One User Class (2D Markov process)
In [24], they consider fixed message length and one user class in the FSMC model but they don’t consider message is variable message length. Each message has independent and identically distributed exponential service duration with the departure rate as follows [20]
where μ depends on k because the departure rate in M /M /∞ queueing mode is proportional to the number of interfering users.
Because the one message is equal to variable number ( on average) of fixed length packets, the message arrival rate λ (messages/sec) is related to the average number of generated packets within a packet duration G (packets/packet duration) by the following simple conversion formula [17][20]
The state transition diagram for M/M/∞queuing model is shown in Fig. 1.
Fig. 1.The state transition diagram for M/M/∞ queueing model
Thus the M /M /∞ queueing model with the following rate:
where λk and μk is the arrival rate and departure rate in state k , which is the state of the M/M/∞ queueing model, respectively.
Then, the steady state probabilities Pk,j is given by
Like [16], we assume that the interference level is constant during a bit duration Δt .We also assume the interference levels associated with consecutive symbols are neighboring states. The number of interfering packets will increase, decrease or remain the same during the bit duration Δt . Therefore, suppose that at the i-th bit in a packet, there are k interfering packets, there are possible k-1, k, k+1 interference packets at the (i-1)-th bit. Different from the previous with throughput analysis in AWGN channels, the DS CDMA/unslotted ALOHA system in Rayleigh fading we proposed, has an additional parameter - channel state j. The channel states of the FSMC will downward across, upward across or remain the same during the bit duration Δt . Therefore, suppose that the channel state is Aj at the i-th bit in a packet, , the state is possible Aj-1, Aj, Aj+1 at the (i-1)-th bit according to the transition probability hi,j. The state transition diagram of the proposed system is two-dimensional and shown in Fig. 2. Note that the bit durationΔt is very small due to high data rate, so the probability having more than one interferer added from one symbol duration to the next is close to zero.
Fig. 2.The state transition diagram of DS CDMA/unslotted ALOHA system for variable message length and one user class with Rayleigh fading FSMC model
Define Ps(k, j, i) as the probability of the following: the number interfering packets becomes k on the i-th bit, the fading channel is in the state j on the i-th bit and the packet transmitted successfully from the first bit to the (i-1)-th bit. According, the state transits during the bit duration Δt and the Ps(k, j, i) is calculated recursively as follows:
The initial condition is given by
Therefore, the packet success probability Q is given as following:
Then, the throughput S is given as follows:
4.3 Proposed Throughput Analysis for Variable Message Length and Two User Classes (3D Markov Process)
Now we further divide users into two user classes. k1 is number of interference of the i-th bit in class 1, k2 is number of interference of the i-th bit in class 2 and Ps,n(e,kn,j,i) is probability of class n. The departure rates of class 1 and 2 messages are derived as:
Also, the arrival rate of class messages is obtained as:
where is average number of packets in a class n message.
Define Ps‘(e,k1,k2,j,i) as the probability of the following: the number of interfering packets become k1 and k2 on the i-th bit, the SNR of FSMC fading channel is in state j on the i-th bit and the first (i-1) bits has e bit errors in the transmitted packet. Ps‘(e,k1,k2,j,i) is calculated recursively as follows:
The first five terms in (22) are the probabilities that the first (i-2)th bits havee-1 bit errors and the (i-1)th bit is also in error. The other terms in (22)are the probabilities that the first (i -2) th bits have e bit errors and the (i-1)th bit is correct.
The initial condition is given by:
where
Therefore, the packet success probability Q’ is given as following:
where t is the maximum number of correctable bits per packet by using error correction codes.
Fig. 3.The state transition diagram of DS CDMA/unslotted ALOHA system for variable message length and two user classes with Rayleigh fading FSMC model
Then, the throughput of two user classes S is given as follows:
where (L-2t)/L is the code rate of the error correction code
where Sn is throughput of class n, Stotal is throughput of whole system.
The normalized (per unit bandwidth) throughput of two user classes Snorm is given as following:
where N is the spreading codes length and also the bandwidth expansion factor, and L/Tp is to convert packets/packet duration to bits/s because one packet has L bits and the packet duration is Tp.
5. The Simulation And Numerical Result
We simulate DS CDMA/unslotted ALOHA system using binary phase-shift keying (BPSK) modulation, spreading code length (processing gain)N=30,the packet length L = 512 bits, the data rate R = 9.6 k bps. We set average length of messages is 5 for one user classes, same as [17]. Then, we set the average length of class 1 messages is 7 and class 2 messages is 5 for two user classes, same as [18][19]. The number of SNR states of FSMC channel model is 8. The maximum Doppler frequency shift Fd is 10, 50 and 100 Hz. Because FSMC state usually over two, we chose 8. According to [32], they analyze 3 to 10 states for different modulations so we chose the 8 for our system. In Figs. 4-12, we use error correction code, and the number of correctable bits is 6, 8 and 10.
Figs. 4-6 shows that the simulation and numerical result of the proposed model versus the various Doppler frequency shift Fd with the number of correctable bit and 10, respectively. They first show that the throughput decreases as the Doppler frequency shift increases if the number of correctable bit is fixed. We can observe another thing comparing Figs. 4-6: for Fd=10, the peak throughput is 4.5, 5.2, 6.0, for respectively; for Fd=50, the peak throughput is 3.3, 4.0, 4.8, for respectively; for Fd=100, the peak throughput is 2.7, 3.4, 4.2 for respectively. Therefore, the throughput increases as the number of correctable bit increases if the Doppler frequency shift is fixed. This is because error correction codes convert some unsuccessful packet transmission in the successful ones. We also observe that the system throughput decreases as Doppler frequency shift Fd increases. This is because the probability of the received SNR state crosses down the poor SNR state increases as Doppler frequency shift Fd increases.
Fig. 4.The throughput versus SNR=10, Fd=10, 50 and 100, t=6 with variable message length for one user class
Fig. 5.The throughput versus SNR=10, Fd=10, 50 and 100, t=8 with variable message length for one user class
Fig. 6.The throughput versus SNR=10, Fd=10, 50 and 100, t=10 with variable message length for one user class
Figs. 7-9 are DS/CDMA unslotted ALOHA with variable message length for two user classes in Rayleigh fading FSMC model without access control. That is Ptr,n = 1. In Fig. 7, for Fd=10, when message arrival per packet duration over 1.0, system throughput will decrease from the peak throughput 5.9. We find similar phenomena in different Doppler frequency offsets in Figs. 8-9. In Fig. 8, for Fd=50, when message arrival per packet duration over 0.9, system throughput will decrease from the peak throughput 4.7. In Fig. 9, for Fd=100, when message arrival per packet duration over 0.8, system throughput will decrease from the peak throughput 4.2.
Fig. 7.Simulation and numerical result of SNR=15dB without access control, Fd=10Hz, t=6 with variable message length for two user classes
Fig. 8.Simulation and numerical result of SNR=15dB without access control, Fd=50Hz, t=6 with variable message length for two user classes
Fig. 9.Simulation and numerical result of SNR=15dB without access control, Fd=100Hz, t=6 with variable message length for two user classes
The access control in (10) can maintain total throughout at maximum when high message arrival, as shown in Figs. 10-12. For example, in Fig. 10, for Fd=10, when message arrival per packet duration is over 1.0, the total throughput remains at the peak 5.9 when message arrival per packet duration is over 1.0. For comparison, in Fig. 7, the total throughput drops from the peak 5.9 when message arrival per packet duration is over 1.0..
Fig. 10.Simulation and numerical result of SNR=15dB with access control, Fd=10Hz, t=6 with variable message length for two user classes
Fig. 11.Simulation and numerical result of SNR=15dB with access control, Fd=50Hz, t=6with variable message length for two user classes
Fig. 12.Simulation and numerical result of SNR=15dB with access control, Fd=100Hz, t=6with variable message length for two user classes
For each simulation point in Fig. 4-12, the number of packets simulated is 500,000. We can see the simulation and numerical results are very close, so the number of packets simulated is sufficient to ensure high degree of confidence level.
6. Conclusion
In previous papers, they mostly consider the non-fading channels. In the papers[21][22][23], the authors consider only that the channel coefficient of Rayleigh fading was the same in the whole packet, which is not realistic. In [24], we proposed the FSMC to analyze throughput performance of the uncoded DS CDMA unslotted ALOHA networks with fixed length data traffic and one user class. The channel gain as well as multiple access interference can vary from symbol by symbol. In this paper, we propose the FSMC to analyze throughput performance of the coded DS CDMA unslotted ALOHA systems for variable message length with one or two user classes, and access control in mobile channels. Compared to [24], we add channel coding, variable length data traffic, and two user classes in this paper. In this paper, we analyze the effect of channel coefficients and the effect of Doppler frequency shift symbol-by-symbols. We can observe that the system throughput decreases as the Fd increases. The proposed use of FSMC allows us to model the symbol-by symbol SNR variation in mobile channels. The numerical and simulation results both show that our proposed two user class network can reach maximum system throughput in high offered load.
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