Steady State Analysis of Base Station Buffer Occupancy in a Cellular Mobile System

Te 3rd generation partnership project (3GPP) standards organizations makes great eforts in order to reduce the latency of 5G mobile networks to the least possible extent. Recently, these networks are associated with big bufers to maximize the network utilization and minimize the wasted wireless resources. However, in existence of the TCP congestions, having bottlenecks are still expected on radio access networks (RANs) data paths. Consequently, this infuences the network performance and reduces its quality of services (QoSs). Apparently, studying and improving the behavior of bufers deployed at 5G mobile networks devices can contribute to solving these problems (at least by reducing the queuing time at these bufers). In this paper, we study the bufer behavior of base stations in a 5G mobile network at steady state. We consider a cellular mobile network consisting of fnite number of users (stations, terminals, and mobiles). At any time-slot, a station may be using the channel (busy) or not using the channel (idle). Since system analysis of cellular mobile networks in general form is rather complex, solutions are always obtained in closed forms or by numerical techniques. A two-dimensional trafc system for cellular mobile networks is presented, and the main performance evaluations are derived. Moreover, diferent moments of the base station bufer occupancy are calculated. Te study reveals that there is a correlation between the state of the mobile stations (busy or idle) and the expected bufers occupancy of the base station. In addition, the results discussions demonstrate some important factors and parameters that afect the base station bufers and the overall network performance. Tese factors can be further worked on and controlled to obtain the least possible latency in next generation mobile networks.


Introduction
Bufer management is such an important network parameter that afects the quality of service of data trafc. In the study of [1], bufer sizing in wireless networks has been studied addressing the unique challenges of wireless environments such as time-varying channel capacity, variable packet interservice time, and packet aggregation. Tey classifed the current state-of-the-art solutions, discuss their limitations, and provide directions for future research in the area. Furthermore, wireless sensor network (WSN) has emerged as the new technology that will have a profound efect in all the felds being wireless in nature. Data packet delivery process in WSN was discussed in [2] with the help of two bufer policies. Because two diferent priorities (high priority and low priority) are applied at each node. Te number of packets to be transmitted by the nodes in the route is decided with two bufer policies, which is single bufer policy and dual bufer policy. Cellular mobile networks have been afected signifcantly by the concept of software-defned networking (SDN). Te type and the capacity of output bufer, which stores packets temporary, have infuenced mainly the average service time of an OpenFlow switch. Reference [3] modeled the handover delay due to the exchange of OpenFlow-related messages in mobile SDN networks. Te total delay encountered by a mobile node while in a handover process, to establish a session from the switch in the source eNodeB to the switch in the destination eNodeB, is called the handover delay. Moreover, the study of [4] presented steady state analysis of bufer occupancy for diferent forwarding strategies in mobile opportunistic network (MON). Actually, depending on local information exchange to measure bufer occupancy in bufer management in MON had brought overhead. Consequently, to fnd the mean bufer occupancy, it is better to study the aggregated bulk transfer size using real-life contact traces and fnd that it follows a log-normal distribution. However, results of this paper help in measuring how fast a node bufer gets depleted when applying diferent routing algorithms. Tus, helping in designing better bufer management techniques and routing algorithms.
Recently, great attention has been paid to the mobile services especially in cellular systems which has covered urban areas. A lot of topics concerning these systems have been studied, i.e., frequency assignment techniques, channel access methods, transmission quality, standards for interfering with the wired networks, and trafc analysis. Considering the last topic, many performance measures of voice systems have been evaluated with mathematical modelling. Asynchronous time division multiplexing (ATDM) scheme is used for transmitting packets coming from many users on a single channel simultaneously. While waiting for transmissions on the channel, the data packets are stored in the ATDM bufers (statistical multiplexer). Te aim of this study is investigating the bufer behavior of random-multiple access base station and cellular mobile networks. Tese systems are characterized by the fact that a number of mobile stations exchange digital information by using a distributed random access algorithm on a common radio channel. Whenever a given station attempts transmission of a packet to another station, the attempt may be unsuccessful, in which case the packet should be retransmitted. Unsuccessful transmission may occur due to the channel noise, or because of the interfering from another station trying to send a packet over the common channel at the same time, or because the intended receiver is itself in a mode of transmission. Data packets coming from diferent stations can share a single communication channel through asynchronous time division multiplexing (ATDM) system (or statistical multiplexer [5]). All packets waiting for service are temporarily stored at the bufer of the statistical multiplexer.
Tis is the organization of the rest of this paper. Section 2 presents the used mathematical model and the main model assumptions. Section 3 introduces the base station bufer analysis at the steady state and the corresponding probability generating function (PGF) is derived. In Section 4, mean base station bufer occupancy at steady state is calculated. Section 5 introduces discussion and comments on the results. Section 6 concludes the study.

Mathematical Model
We consider a cellular mobile network with k independent and identical stations (sources, terminals, . . .). Data generated by diferent stations are divided to small fxed size packets and saved in the base station bufer. Packets can be transmitted from the bufer only at the beginning of each slot. Each station alternates between two independent states with arbitrary length: state of transmission (busy) and a state of not transmitting (idle). So we have the following: λ: probability that a busy station in a given slot will remain busy in the next slot. 1 − λ: probability that a busy station in a given slot will become idle in the next slot. μ: probability that an idle station in a given slot will remain idle in the next slot. 1 − μ: probability that an idle station in a given slot will become busy in the next slot. where λ + μ ≠ 1. Actually, this helps to add a type of correlation between diferent stations. During each slot, a busy station generates a number of packets N with PGF N(z), where this function is independent from one busy station to another. N(z) can be proposed so as to add diferent levels of the activity of the station.
Let the random variable (RV) c l represents the number of busy stations during slot. It is obvious that both busy and idle states of the k stations have geometric distributions. So the value of c l+1 can be obtained from c l , as follows: where c l j�1 A j specifes how many stations are busy in slot l will remain busy in slot l + 1, and k− c l j�1 B j specifes how many idle stations in slot l will change to be busy in slot l + 1 i.e., Note that, A j , B j are all Bernoulli RVs, where A 1 � 1, if the frst busy station in slot l will remain busy in slot l + 1. A 1 � 0, if the frst busy station in slot l will change to idle in slot l + 1. Te same is applied to other busy stations using RVs A 2 , A 3 , . . . , A c l . And, B 1 � 1, if the frst idle station in slot l will remain idle in slot l + 1.
Terefore, the group of RVs A j ′ s and B j ′ s can be considered as a group of independent and identically distributed Bernoulli RVs with common PGFs A(z), B(z), respectively. Here, If the number of packets entering the bufer during slot l is represented by the RV D l , hence where frst busy station generates N 1l packets, second busy station generates N 2l packets, and so on. Tese RVs are independent and identically distributed with common PGF N(z). Now, let the number of packets stored in the base station bufer at the beginning of slot l + 1 be denoted by the RV w l , then we have where and D l+1 represents the number of packets entering the base station bufer during slot l + 1.

Steady-State Buffer Analysis
It is obvious from equation (7) that the value of w l+1 is not dependent only on w l , but rather on D l+1 also. However, since D l+1 is dependent on c l+1 (from equation (6)), we assume that after a long time (as l ⟶ ∞) the distribution of the system state in an arbitrary slot no longer varies with time and we use a two-dimensional Markov chain that describes the base station bufer in terms of the pair (c l , w l ). Let S l (x, z) represents the joint PGF of c l , w l , so Ten, Using equation (7) in equation (9), then Using D l+1 from equation (6) in equation (12), hence which can be written in the form from which we can obtain Using c l+1 from equation (1), yields where A j and B j are all i.i.d RVs with common distribution, where Substituting in equation (17), hence

Journal of Computer Networks and Communications
that can be written in the following form: (20) However, since a busy station generates at least one packet that cannot leave the bufer before the next slot, the last expression can be written in the following form: At the steady state, S l (x, z) and S l+1 , which are the joint PGFs of the number of busy stations and the number of packets saved in the base station bufer, will converge to S(x, z).
In view of (4) and (5), (22), gives which can be written in the following form: If S 0 is the probability of an empty bufer (S 0 � Pr [w � 0]), then S(x, z) should satisfy the following:

Journal of Computer Networks and Communications
Although no explicit formula for S(x, z) can be obtained, we can derive many results from equation (25) considering that where C(x) is the PGF of the number of busy stations, and where L(z) is the PGF of the base station bufer occupancy at the steady state. Substituting for z � 1 in equation (25) we can get an expression for C(x) However, an explicit formula for C(x) (which represents the number of busy stations at steady state) can be obtained equation (30) knowing that it is a polynomial of degree k, to get Substituting for C(x) from equation (31) in equation (30), gives which gives a system of (k + 1) equations in the (k + 1) unknown c 0 , c 1 , . . . , c k . Now, let us focus on a specifc station of the k stations where the average length of the busy period of this station is 1/1 − λ and the average length of the idle period is 1/1 − μ, then the probability that this station is busy during any selected slot, is given by Let Ten, a specifc station is busy with probability ω and idle with probability (1 − ω). Considering one station i, let v i be Bernoulli RV represents the number of busy stations (0 or 1), where Previous relation is applied to all k stations. Since all stations are identical and independent and so are the RVs v i . Terefore, C(x) (the PGF of the total number of busy stations) is given by But we have Journal of Computer Networks and Communications So, C(x) can be written as Next, we turn the attention to the steady state distribution of the bufer occupancy. Equation (25) can lead us to the following relation Equation (39) is a quadratic equation which has two roots for x in terms of z. One of these roots satisfes that x � 1 for z � 1. Consider x � p(z) is that root of equation (39). Substituting for this root in equation (39), yields When x � p(z) in equation (25), gives which can be written as

Mean Base Station Buffer Occupancy
Although equation (50) does not give an explicit formula for the generating function of the base station bufer occupancy, many steady-state features of the bufer can be derived from it. Te base station mean bufer occupancy G at the steady Equation (50) leads us to From equation (38), we fnd that When p(z) � 1, then Substituting from equation (54) in equation (52), hence Now, we proceed to get the mean bufer occupancy G of the base station from the relation 1).

Journal of Computer Networks and Communications
Before applying L'hospital rule on equation (50), let us consider the following (59)

So, S(p(z), z) is written as
where Now, the desired derivative becomes Substituting for the values of u ′ (1), v ′ (1), u ″ (1), and v ″ (1), we conclude 8 Journal of Computer Networks and Communications .

(63)
Second, to fnd p ′ (1), from equation (40), we have Taking the frst derivative with respect to z, then Substituting for z � 1, we get Te previous result has been approved using Mathematica program [6] in calculating p ′ (1). Solving for p ′ (z), therefore from which we fnd Using equations (63) and (54) in equation (56), we get After using the value of p ′ (1) from equation (68) in equation (69), then Solving for G, then where the values of M ′ (1) and M ″ (1) can be obtained from equation (45) in terms of known parameters on one hand and the derivatives of p(z) at z � 1 on the other hand. From equation (45), we have which agrees with the result of Mathematica program when used to calculate M ′ (1). Substituting for the value of p ′ (1) from equation (68) in equation (73), we get which has been approved with the result of Mathematica. Now, to fnd M ″ (1), we proceed as follows: which gives (the last result was verifed again using Mathematica program). Now, we need to fnd the value of p ″ (1), from equation (65), and using Mathematica for simplicity, we will get Substituting for z � 1, then which gives Simplifcation of the last relation with Mathematica, we get Using Mathematica to simplify the previous equation, then Higher moments of the bufer occupancy can also be obtained, using the same way; however, this is going to lead to complicated mathematical derivations.

Discussion of the Result
Te obtained results for the steady-state distribution of the number of busy stations shows that it depends only on the value of the parameter ω. Tis result may lead us to say that the steady-state bufer behavior of the base station is determined only from the value of ω. Tis section will be used to discuss this point. We consider that each station is busy with probability ω and is idle with probability 1 − ω, independently from slot to slot. So mean number of busy slots � 1 (1 − ω) .
In such case, the average activity of the station will also be ω if the mean numbers of busy and idle slots are both multiplied by a same factor l, i.e., if λ and μ are selected, such that mean busy slots To demonstrate the importance of the parameter l, let us use the following situation. Suppose that the busy stations, every busy slot, generate one message per busy slot. Terefore, the number of packets generated by the busy stations equal to the message length (in packets). Assuming a geometric distribution for the message length and assuming that the random variable N represents the number of packets generated by a station in a given slot (message length), we then get where ψ is the probability that the message not fnished and (1 − ψ) is the probability that the message is fnished, then 12 Journal of Computer Networks and Communications where N is the mean message length. In such case, the mean bufer occupancy of the base station, at the steady state, can be obtained in the form (93)

Conclusion
Te study and analysis of base stations bufers behaviors in 5G and next generations mobile networks can contribute to reducing the network latency and improving the network performance and the QoS. In this paper, the bufer behavior of base stations of 5G mobile networks at steady state is investigated. Te network includes a base station and a fnite number of mobile stations. Each mobile station alternates between two independent states with arbitrary length: state of transmission (busy) and a state of no transmission (idle). A two-dimensional Markov chain has been used to derive the probability generating function corresponding to bufer occupancy at the steady state. Mean bufer occupancy of the base station of the cellular mobile network at the steady state is also calculated. Te results show a type of dependency between the activity level of the mobile stations (busy or idle) and the expected bufer occupancy of the base station. Moreover, expressions resulted from the analysis have listed factors and parameters that afect the base stations bufer behavior. Tese factors can be studied and analyzed to further reduce the latency and improve the QoS of next generation mobile networks.

Data Availability
No underlying data were collected or produced in this study.

Conflicts of Interest
Te author declares that there are no conficts of interest.