Model and analysis

We proceed to model the evolution of Message 4 buffer, with a few more as-sumptions, taking into account all the four steps described in Section 2.3.2 using a two-dimensional Markov chain. This model can then be used to determine the maximum throughput θmaxof the random access requests

Machine-to-machine random access

that a system can sustain. We define a timeslot¹as the duration of one random access opportunity that consists of b subframes. One preamble corresponds to one Slotted Aloha channel which can be accessed only at the beginning (the first subframe) of a timeslot. We assume that there are K preambles which are randomly selected by the UEs, making the first step of the random access process effectively a Multichannel Slotted Aloha (MCSA) system with K parallel channels. Every preamble which has been used by just one UE is referred to as a successful preamble while the ones used by at least one UE are referred to as the chosen preambles.

Their numbers in timeslot n are denoted by Y_n⁽¹⁾and ˜Y_n⁽¹⁾, respectively and obviously Y_n⁽¹⁾≤ ˜Y_n⁽¹⁾ ≤ K. Furthermore, we assume that the fresh requests arrive according to a Poisson process with λ arrivals per sub-frame. Although we do not explicitly model the backlog mechanism, the aggregate random access requests, which comprises of both the fresh and the retransmitted requests, is assumed to form a Poisson stream²of rate aarrivals per subframe.

In Step 2, we consider that c uplink (UL) grants can be provided in a Message 2 which consumes N^Msg2Control Channel Elements (CCEs) in PDCCH. Since these messages/UL grants are not queued, they are lost if not sent in the timeslot immediately following the timeslot when Mes-sage 1 was sent because of small value of c. Failure to do so fails the random access process, and a backoff period begins before a retransmis-sion attempt is made. Again, analogously as in Step 1, we denote by Y_n⁽²⁾ and ˜Y_n⁽²⁾the numbers of the successful and the chosen UL grants, respec-tively, sent in timeslot n. In a timeslot, at most bc UL grants can be sent.

If ˜Y_n−1⁽¹⁾ ≤ bc then we do not have any loss in Step 2, i.e., ˜Y_n⁽²⁾= ˜Y_n−1⁽¹⁾ and Y_n⁽²⁾= Y_n−1⁽¹⁾. On the other hand, if ˜Y_n−1⁽¹⁾ > bc, then ˜Y_n⁽²⁾= bcand the UL grants are sent corresponding to the randomly selected chosen preambles.

In Step 3, all the users that received a Message 2 corresponding to their chosen preamble send a Message 3. Thus, the number of successful Mes-sage 3 in timeslot n is Y_n⁽³⁾= Y_n−1⁽²⁾.

A Message 4 is generated by the eNB for every successful Message 3 received from the previous timeslot. Each Message 4 uses N^Msg4CCEs.

These messages can be queued, and they should be received by the UE 1Note that this definition of a timeslot is different from the definition of a time slot described in the LTE standards and mentioned at the beginning of Sec-tion 2.3, which is always 0.5 ms long.

2The evidence for the validity of the Poisson assumption in provided in Ap-pendix B of Publication I.

Machine-to-machine random access

X_n+1 Y_n⁽⁴⁾ Y˜n⁽²⁾

X_n Yn⁽³⁾

Y_n−1⁽²⁾ Y_n−2⁽¹⁾

Y˜_n−1⁽¹⁾

Figure 2.3. Random variable dependence graph for the variables in (2.1).

before the expiration of the retransmission timer. Otherwise the random access process fails. Let N be the size of PDCCH resource (in CCEs).

Then a maximum of either M = _NMsg4^N or m =^N−N_NMsg4^Msg2Message 4’s can be sent in a subframe depending on the presence or absence of a Message 2 in the PDCCH. Additionally, if c ≤ m, a Message 4 buffer need not be formed as there is enough capacity in the PDCCH to send a Message 4 corresponding to all the successful Message 3’s. Under this condition, the process fails due to one of two reasons—either there is a collision in Step 1 or there is a loss in Step 2 because c is too small. On the other hand, if c > m, due to the limited capacity of PDCCH, a buffer may have to be used to keep Message 4’s at the eNB before sending them. This may lead to the failure of the random access process, even if the two aforementioned failure events do not happen, due to a long buffering delay of Message 4.

With these assumptions, we observe that the Message 4 buffer evolves as

X_n+1= X_n− Y_n⁽⁴⁾+ Y_n⁽³⁾= X_n− Y_n⁽⁴⁾+ Y_n−1⁽²⁾, with (2.1) Y_n⁽⁴⁾= min{Xn, m}1^Y˜n⁽²⁾>0+ min{Xn, M}1^Y˜n⁽²⁾=0

, (2.2)

where X_n and Y_n⁽⁴⁾ denote the length of the Message 4 buffer and the number of Message 4’s sent in timeslot n, respectively. The pair

X_n, Y_n⁽³⁾ is an aperiodic and irreducible Markov chain. The Markov property can be verified from the dependence tree of different variables as shown in Figure 2.3; aperiodicity and irreducibility follow from the construction of the chain.

Let the average throughputs of Message 1, Message 2 and the total number of uplink grants be denoted by θ¹(a), θ²(a)and ˜θ²(a), respectively.

Furthermore, let the average leftover capacity for serving the Message 4 buffer per timeslot be denoted by σ⁽⁴⁾(a). In Publication I, § V, we de-scribe in detail the method to calculate these quantities from the system parameter values. As the main theoretical contribution, we prove that the Message 4 buffer is stable if and only if

θ⁽²⁾(a) < σ⁽⁴⁾(a)

Machine-to-machine random access

in Appendix B of Publication I. That is, the Message 4 buffer is stable if and only if the throughput of Message 2’s, which is sent in the PDCCH and possibly generate Message 4’s, is less than the capacity of sending Message 4’s in the PDCCH. Furthermore, let a^∗₂and a^∗₄represent the max-imum arrival rate at which the Message 2’s and Message 4’s can be sus-tained, respectively, and θ^∗= θ⁽²⁾(a^∗₄)be the maximum throughput of the random access requests. Finally, we reiterate that the input parameter of our model is the aggregate arrival rate a. However, when the system is stable, the average arrival rate λ of the fresh requests is the same as the average throughput θ⁽²⁾(a)of Message 2’s. Therefore, the relation

λ(a) = θ⁽²⁾(a), (2.3)

can be used to infer the arrival rate of fresh requests from the knowledge of the average throughput.