Electronic Design

Modeling Mazimizes Wireless Telephone Network's Effectiveness

Though well established from a technical standpoint, wireless telephony still faces numerous obstacles on its path to a ubiquitous, satisfying user experience. Still, the best way to acquire more subscribers and keep them satisfied is to make the service as easy to use and as reliable as possible. Wireless companies should strive to become equal with land-based connections in terms of voice quality, pervasiveness of connections, and value-added services.

Unfortunately, this level of service isn't so easy to achieve. At the core of the problem is the fact that wireless basestations can only handle a finite number of calls. Until they ramp up in capacity, there will always be situations where users cannot make a connection. If these users are mobile, any connections that are established can be easily dropped during handoff.

These problems are only exacerbated by challenges such as public gatherings, sporting and concert events, traffic congestion, and the notoriously unpredictable, accident-related basestation overloads. To alleviate these difficulties, the art of traffic engineering has taken center stage in the drive to improve the user experience.

Traffic engineering is key to effective wireless telephony network design and planning. In the process, traffic characteristics need to be addressed. Once accomplished, mathematical models can be applied to the dimensioning of the network. Dimensioning determines the amount of traffic the radio equipment captures. An accurate traffic model for the system greatly enhances the accuracy of network dimensioning, extending network investments and increasing the profit returns. Because the equipment is used more efficiently, it can capture the majority of the traffic and produce greater user satisfaction.

On the flip side, poor modeling of traffic characteristics can actually affect system performance. When cells are under-dimensioned, not enough radio channels are installed. This congests the cell, and it also may affect overall system performance.

Before approaching the problem of network dimensioning, it's essential to understand the Erlang-B formula, which forms the basis of the required mathematical models. Once understood, the formula can be used to model the various real-life scenarios, providing accurate dimensioning of any given network.

Traffic is defined as the use of given resources. Traffic on a highway is the number of cars that use the highway for transportation. Increased traffic on the highway implies that it is used more1. Bank traffic involves the use of tellers. Greater traffic on the system implies a better use of tellers. In wireless telephony networks, traffic is defined as the use of radio channels.

Revenue from a wireless network is directly proportional to the amount of time that communication channels are being occupied. When a user makes a phone call, a channel is seized for communications, generating traffic. As a result, traffic is proportional to the average call duration.

Traffic is related to the use of radio channels. Traffic value is directly proportional to the frequency of phone calls and the average duration of those calls. Traffic per subscriber (ρ′) in a wireless network is defined as

ρ′ = λ′ × 1/µ Erlang

where λ′ is the number of calls a user makes in a given period, and 1/µ is the average duration of each call. The unit is the Erlang2. The equation can be interpreted as how long on average a user occupies a radio channel, within a certain amount of time. In traffic-engineering terminology, traffic per subscriber is often called traffic intensity. If a user in a wireless system makes an average of two phone calls an hour, with each call lasting 1.5 min., the traffic per subscriber (ρ′) would be

ρ′ = 2/60 × 1.5 = 50 mE

Users in a telephony network on average would call n times an hour. Each call would last for m seconds. Traffic intensity, or traffic per subscriber (ρ′), would therefore be

ρ′ = (n × m)/3600 E      (1)

A wireless network can be generally modeled as a queuing system. A queuing system has customers, servers, and waiting rooms. Its characteristics are governed by the arrival behavior and serving behavior. Arrival behavior is the probability distribution of customer arrivals—that is, how often a customer would arrive at the system. Serving behavior is the probability distribution of service. Upon arrival, customers will spend time being served. The distribution of the serving time is the serving behavior. In queuing theory, different models are based on different queuing scenarios. The simplest one takes the form M/M/1/×.

The first M, which represents the arrival distribution, is memoryless3. The arrival of one customer is independent of other customers. This is generally true for any wireless communication system. The second M, representing the serving distribution, also is memoryless. The serving time of a customer is independent of all other customers. The 1 means there is only one server in a system, while the × indicates that there is an infinite number of waiting rooms in the queuing system4. The probability of this system having k customers (Pk) is defined as

Pk = (1 − ρ)ρk

where ρ = λ/µ′ is the total arrival rate, and 1/µ is the mean serving time.\[1\]

Engineers using this model make four assumptions. Primarily, the population of customers is infinite. That is, there is an unlimited supply of customers. Next, a served customer leaving the system would not immediately return. Third, all customers waiting in the line cannot leave the system. Finally, the system would be first-come, first-served.

A similar model can be applied to wireless telephony systems. Servers are radio traffic channels in each cell. Customers are calls from different mobiles. In queuing theory's terminology, a cell with n radio traffic channels can be described by M/M/n/n. Erlang studied this system and devised some very useful formulas during the 1910s. His research lead to the Erlang-B formula, which is the most widely used formula for traffic engineering.

The queuing model of an M/M/n/n system can be represented by Figure 1. In the diagram, λ is the total arrival rate, which is different than that defined in traffic intensity. It also is defined as the rate at which customers from a group of customers arrive at a system. Meanwhile, n is the number of servers in the system. Each customer will be served at the rate of µ, meaning each customer will finish service in 1/µ unit of time. Distributions of customer arrivals and service times are assumed to be memoryless. There is no waiting room at the system. When all n servers are busy, customers are lost. When a customer has finished service, it leaves the system forever and doesn't return to the system for additional service.

Arriving And Leaving Customers
Customers arriving and leaving the M/M/n/n system can be described by a state diagram (Fig. 2). There are n + 1 possible states that represent the number of customers in the system. The rate of the system changing from state i to i + 1 for i = 0, 1, ..., n − 1 is λ—i.e., when a new customer comes to the system and finds an available server.

The rate of the system changing from j to j − 1 for j = 1, ..., n is jµ, meaning a customer has finished service and leaves the system. State j changes to state j − 1, related to j. When j customers are being served in the system, the probability of a customer leaving a system would be j times higher because any one of the j customers finishing service would change the state of the system.

The flow between states in a stable system would be equal. Otherwise, the system would either be congested all the time or empty all the time. The equilibrium's validity is proved in Queuing Systems: Volume 1, Theory. With this equilibrium property,

λPi = (i + 1)µPi + 1, for i = 0, 1, ..., n − 1

where Pi for i = 0, 1, ..., n are the probabilities of the system having i customers. Additionally, the property of sum of all possible states' probabilities would equal 1, or

The probability of k customers in the system can be obtained from

where ρ = λ/µ and k = 0, 1, ..., n.

Here, ρ is defined as the offered traffic or offered load. Since λ is the total arrival rate, ρ is the total offered load from the customers. As the probabilities for a system having k customers for k = 0, 1, ..., n are obtained, designers will wonder when they will start to see congestion.

After following through, the answer can easily be found. When all n servers are busy, a new customer coming to the system is blocked. That's Pn. As a result,

where ρ = λ/µ. This is known as the Erlang-B formula. It's critical because it governs the relationship among blocking probability (PBLOCKING), offered traffic (ρ), and the number of traffic channels (n). Given any two of the values, the remaining variable can be determined. In a wireless telephony network, it's essential to determine how many channels are required to capture the traffic.

Traffic in a wireless telephony network can be modeled by M/M/n/n, where n is the number of traffic channels in a sector. Consequently, traffic value can be calculated by the Erlang-B formula. Given the quality of service (QoS), which is equivalent to the blocking probability (QoS = PBLOCKING), and the number of radio channels in a cell (n), we can obtain the offered traffic by using Equation 2. Furthermore, the actual captured traffic (^ρ) can be calculated by

^ρ = ρ(1 − PBLOCKING)      (3)

The four useful values—QoS, number of radio channels (n), offered traffic (ρ), and captured traffic (^ρ)—are all governed by Equation 2 and Equation 3. All of the traffic characteristics can be obtained with these two equations. Whenever two of the four values are defined, the other two can be calculated accordingly. Finally, engineers should relate offered traffic to the total number of subscribers in a cell through

This equation is derived simply by using the direct proportionality property of the offered load to the number of users accommodated in a cell.

The variables in the Erlang-B formula, or the assumptions, are affected differently in various scenarios.

Even distribution—the ideal case—is the simplest model used in traffic engineering. It assumes that the distribution of users would be equal, as each of the cells would capture the same amount of traffic. Prior to the availability of radio-planning tools, engineers would use this assumption to estimate the system capacity. Its primary use is to generate an estimation of network capacity. Designers should not base the dimensioning of a network on this.

Distribution based on land usage is a more realistic model. Traffic distribution is highly related to the way the land is used. For example, there would be more traffic in a built-up area or at an airport and less traffic in rural areas. As a result, engineers must weigh the traffic from different land usage when characterizing traffic distribution. This is pretty easy with the help of a sophisticated radio-planning tool, such as Logica Odyssey.

First, designers have to define the amount of traffic in an area. Then, traffic can be spread over the area with weighting based on different types of clutter (land usage). The result of spreading defined traffic based on different clutter-type usage is shown in Figure 3. Figure 4 reveals the corresponding clutter type based on land-usage characteristics.

Weighting Enhances Accuracy
Studies have proved that weighting greatly enhances traffic-distribution accuracy. Yet the figures used in weighting need to be tuned to reflect the characteristics of user behavior and distribution. Engineers can use live traffic data and Odyssey to accomplish this. If no live data is available, engineers must employ rough estimation.

The modeling of highway telephony traffic is much different from typical telephony traffic. Highway traffic is dynamic, as cars are assumed to be moving most of the time. The average duration of a call served by a cell has to be estimated based on the serving distance (the length of highway that the car is on) against the average speed of a car driving on the highway. This complicates the model.

With live traffic data, engineers can scale the traffic per subscriber for stationary users using a constant factor. This factor can be obtained by tuning the length of highway along the highway. Odyssey can spread traffic along a highway, which helps traffic engineers figure out the distribution of highway telephony traffic.

Despite this, traffic distribution is totally different when the highway is congested. Experience shows that when traffic accidents happen, the cell covering that area is congested immediately. The traffic is more concentrated when the users are waiting. Also, they tend to make more phone calls to explain that they are in traffic. Unless a consistent congestion is seen every day, it's almost impossible to consider cases where accidents happen. Extra radio channels can be installed for areas where traffic accidents are more likely to occur.

Traffic mobility is extremely difficult to estimate. But it can be accommodated by reserving channels for traffic transfer from neighboring cells. When a user makes a phone call in a moving car, the call is handed over from one cell to another. Traffic is transferred from the cell where the call was set up to the other cells.

Modeling this traffic behavior is extremely difficult unless the mobility of the users is determined. In the traffic characterization and modeling mentioned earlier, all users are actually assumed to be stationary, or they would be served by the same cells in the call. Engineers, then, have the challenge of modeling moving traffic.

Fortunately, statistics show that traffic in commercial and residential areas is not as dynamic. Most of the time, users in those areas don't move too fast, or they may even be stationary. When it comes to moving traffic, engineers should only be concerned with highway traffic because of its rapid speeds. A call is usually served by a number of cells. Then, engineers must compute how this affects the traffic modeling. To model this kind of traffic, engineers have to obtain some basic information about the mobility behavior.

Looking at the traffic in a cell covering a certain length (l) of a highway, engineers first have to know how many cars drive through the coverage area during the busiest hour. This quantity is defined as x. Next, they have to find out what percentage of those cars driving through the coverage area are network subscribers. This quantity is defined as p%. Third, the average speed of the cars, defined as v, must be determined. The average call duration is t seconds, while each user would make n calls an hour. It is further assumed that those cars are traveling at about the same speed and that they are all homogeneous.

The time to drive through the coverage area is equal to l/v. Traffic flow through the cell is therefore

Here, t >> l/v is assumed. This is the worst-case scenario. Calls are assumed to be on all the time when users drive through the coverage area. From the calculation, traffic on the highway is lighter than the stationary traffic.

This simple formula provides an idea of how to model offered traffic for moving subscribers. But as the equation states, there are a number of assumptions. Before this model can be applied to real-case scenarios, it needs to be refined to describe the scenario as accurately as possible. For instance, even though a given cell might be assigned to cover a highway, it might pick up extra traffic from surrounding urban or downtown areas.

Furthermore, heterogeneous moving traffic complicates the modeling. When a cell is covering a highway—including an exit on the highway—during rush hour, all of the cars in the slow lane are jammed as they wait to take the exit. Meanwhile, other cars are driving about 55 mph in the fast lane. In this case, traffic modeling must consider the stationary and the moving traffic. When a cell covering a highway gets congested, statistics will show very-high handover failure rates for attempts from its neighbor cells to the congested cell.

Special occasions contribute to a sudden increase of traffic. For some events, like the Super Bowl, NBA finals, and a PGA tournament, the amount of traffic can be estimated in a much easier way.

Estimate the number of people that will be present at the event. Then, estimate the amount of traffic per subscriber. This value would vary due to different calling behaviors at different events. Note that the maximum traffic needs to be estimated. For example, traffic during intermission may be more of a concern than traffic during the game itself. Next, estimate the distribution. Where will the people be concentrated? After that, find out which cells serve the area. Finally, spread the traffic based on distribution, traffic per subscriber, and the number of people.

It's almost impossible to prevent network congestion due to some unexpected events, such as accidents and adverse weather. These cases generate abnormally high traffic. When users experience congestion, they make the problem worse. Upon reaching a busy signal, most people try to make the call again, or even a few times, to get a channel. This can significantly congest the network. Also, it violates the assumption that customers (calls) blocked by the system wouldn't return to the network. The Erlang-B formula, then, doesn't apply to this kind of situation.

A Tough Problem
Unfortunately, no known model can solve the problem. Operators in some countries install more radio channels in cells to cope with this kind of circumstance. For instance, cells covering the airport would have more radio channels installed even when the channels aren't normally used. When adverse weather conditions prohibit most planes from landing and taking off, the engineer can then activate all the radio channels and assign frequencies to them to try to prevent congestion. This can definitely help, but it can't guarantee that congestion won't happen.

Some operators would agree that in these circumstances, users should be discouraged from making calls when they get a busy signal. If they keep redialing, fewer customers will actually get through to make a phone call.

There are some assumptions in formulating the Erlang-B formula. But for a practical wireless telephony network, are those assumptions still valid?

First, there isn't an infinite amount of customers in each sector. Consequently, the blocking probability of a practical network is less than the value obtained from the formula. Besides, the arrival distribution may be distorted. Since the memoryless property can approximate the system and obtain simple results, traffic obtained by the Erlang-B formula may be lower than that in a real situation for a wireless telephony network, due to the infinite-customers assumption.

Second, congestion caused by redialing must be considered. When users experience blocking, they generally redial immediately and try to seize a traffic channel. This further congests the network. When traffic approaches the congestion limit, the blocking probability increases at a much faster rate than the theoretical value. In a real-life scenario, the arrival of customers (when a user makes a phone call) is highly correlated to the customers blocked by the system (when a user gets the busy tone).

Third, for a new wireless telephony network, arrival and serving assumptions are generally referenced to some objective information like population distribution, demographics figures (annual household income), and fixed-line phone installation. This is only a rough estimate of the traffic distribution when the wireless network is first designed. Once the network is up and running, live traffic data generated in the network helps to refine the actual behavior of the mean arrival time and mean service time—hence the traffic amount generated by each subscriber, as well as the offered traffic.

Finally, marketing strategies and pricing schemes greatly affect traffic behavior. When an operator reduces the tariff on mobile calls from 20 cents/min. to 5 cents/min., traffic increases drastically. This would require the addition of more radio channels to the network. Offered traffic and traffic per subscriber increase when subscribers feel comfortable making more calls on their mobile phones. As a result, traffic intensity needs to be revised from time to time in order to reflect real-life scenarios.

Once acquainted with the general idea of traffic modeling, engineers should be able to further elaborate some generic models and apply them to more difficult traffic-behavior instances.


  1. High traffic figures would show high resource utilization. However, congestion would be more of a concern than utilization. A very high traffic count on a highway would show that the highway is well utilized. But it also may mean that all of the cars using the highway would need to spend a long time on the highway due to congestion. Justification on the utilization and quality of service needs to be considered.
  2. The unit Erlang (E) commemorates Danish mathematician A.K. Erlang's contribution to traffic engineering. The intuitive interpretation of the unit is utilization offered to some resources, E = busy time/usable time. In theory, the maximum amount of traffic a channel can capture is 1 E, which means that the channel is busy all the time.
  3. Memoryless distribution is exponentially distributed. That is, interarrival time between customers is exponentially distributed. Exponential distribution was not discussed in detail in this article. For more in-depth study on exponential distribution, see reference 1.
  4. Waiting rooms include those at the servers. If there are three servers and seven waiting rooms in a system, it will be described as M/M/3/10.


  1. Kleinrock, L., Queuing Systems: Volume 1, Theory, John Wiley, New York, 1975.
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