I. Background

Terrestrial propagation modeling has long been used as an experimental arm in wireless communications, in particular, in computer broadband internet cellular networking mobile radio systems. In our previous papers, results we have shown the terrestrial propagation modeling (TPM) using various techniques and methods and parameters particularly for identification of suitable wireless technology[1], then the frequency band management of electromagnetic spectrum in the range of 2.4GHz to 60GHz - comparison of licensed and un-licensed Bands Using the Broadband Wire-Less Internet Networking Cellular Mobile Wi-Fi and Wi-Max Technologies [2], then simulation and modeling paradigms[3], design, planning and implementation [4], TPM using terrestrial propagation mechanisms (TPM) parameters[5], such as, reflection, diffraction and scattering and recently but not lastly and leastly we have shown solving the Maxwell’s equations using propagation mechanisms parameters [6].

In this scientific research technical article, author explain the trunking theory for the trunked radio systems and describing the Erland model B and C formulas rigorously by using mathematical expressions with mathematical theorem provings.

I.1.Trunking theory

Two of the major classes of trunked radio systems are: Lost call cleared (acronym LCC) and Lost call delayed (LCD) systems [7].

Whilst in lost call cleared systems, queuing is not provided for call requests. When a user requests for service, there is a minimal call set up time and the user is given immediate access to a channel if one is available. If all the channels are already in use (or being put-into use) and no new channels are available, the call is blocked without access to the system. The user does not receive service, but is free to try the call again later. Calls are assumed to arrive with a Poisson distribution, and it is further presumed that there are a nearly infinite (∞) number of users. The Erlang B formula describes the grade of service (GoS) as the probability (p) that an arbitrary user will experience a blocked call in a lost call cleared system. It is assumed that all blocked calls are instantly returned to a ∞ user pool, and may be retried at any time in the future. The time between successive calls by a blocked user is a random process and is assumed to be Poisson distributed. This model has long been used as an experimental arm in shipping yard ports. The ships are assumed to be arrived with a Poisson distribution [8]. Hence, this model is accurate for a large system with many channels and many users with similar calling patterns.

Whereas in lost call delayed systems, queues are used to hold call requests that are initially blocked. When a user attempts a call and a channel is not immediately available, the call request may be delayed until a channel becomes available. For LCD systems, it is necessary to find the likelihood that all channels are already in use. Then, given that no channel is available initially, it is necessary to know the probability of how long the call must be delayed before a channel is available for use. The likelihood of a call not having immediate access to a channel in a LCD system is determined by the Erlang C formula. For LCD systems, the GoS is measured by the probability theory that calls will have delays greater than t seconds (> t sec).

The knowledge of the Erlang C formula and the service distribution is used to analyze the GoS. It is assumed that a nearly ∞ number of users are present in the system, and that all calls in the queue ultimately are serviced. The Erlang C model also assumes a large number of channels and a large number of users with similar calling patterns. Typically five or more channels are considered to be a sufficiently large number of channels.

I. (A). Erlang B Model Formula

The Erlang B modelformula determines the probability that a call is blocked, and is a measure of the GOS for a trunked system which provides no queuing for blocked calls (an LCC system). The Erlang B model formula is based up on the following basic assumptions:

Call requests are memory-less, implying that all users, including blocked users, may request a channel at any times.

All free channels are fully available for servicing calls until all channels are occupied.

The probability of a user occupying a channel (called the service time) is exponentially (and/or logarithmically) distributed. Longer calls are likely to happen as described by an exponential distribution (or logarithmic distribution).

There are a finite number of channels available in the trunking pool.

Traffic requests are described by a Poisson distribution which implies exponentially distributed calls inter-arrival times.

Inter-arrival times of call requests are independent of each other.

The number of busy channels are independent of each other.

The number of busy channels is equal to the number of busy users, and the probability of blocking is given as

pdf (blocking) = ---------------- (1)

Where C is the number of trunked channels, and A is the total offered load to the trunked system.

I. A. i. Derivation of Erlang Model B Formula

Consider a system with C channels and U users. Let λ be the total mean call arrival rate per unit time for the entire trunked system (average number of call requests per unit time over all channels and all users), and H be the average call holding time (average duration of a call). If A is the offered load for the trunked system (Ts), then A = λH (i.e., λ=).

The probability (p) that a call requested by user will be blocked is given by

p (blocking) = p(none of the C channels are free) -----(2)

it is assumed that calls arrive according to the Poisson distribution, such that

p= {a(t+τ) – a(t)=n} = (λτ)ⁿ for n = 0,1,2,3,… (3)

where a(t) is the number of call requests (arrivals) that have occurred since t=0 and τ is the call inter-arrival time (the time between successive call requests). The term λ denotes the average number of call requests per unit time over the user population, and is called the arrival time.

The Poisson process implies that the time of the nth call arrival and the inter-arrival times between successive call requests are mutually independent. The inter-arrival times between call requests are exponential and mutually independent, and the probability that the inter-arrival time will be less than some time s is given by

p(τ_n≤s) = 1 - e^-λs, s ≥ 0 --------------------------- (4)

Where τ_n is the inter-arrival time of the nth arrival, and τ_n = t_n+1- t_n, where t_n is the time at which the nth call request arrived. In other words, the probability density function (pdf) for τ_n is [9]

pdf(τ_n) = e ^-λτn, τ_n ≥ 0 ------------ (5)

for every t≥ 0, δ≥ 0,

p {a(t+δ) – a(t) = 0} = 1 –λδ +O(δ) --------------------- (6)

p {a(t+δ) – a(t) = 1} = λδ +O(δ) ------------------------ (7)

p {a(t+δ) – a(t) = 2} = O(δ) ---------------------------- (8)

Where O(δ)is the probability of more than one call request arriving over the time interval δ and is a function of δ such that Lt_δ_à₀{} = 0, where Lt indicates the limit

The probability of n arrivals in δ seconds is found from equation (3)

p={a(t+ δ) – a(t)= t_k} }=(λδ)ⁿ ---------(9)

The user service time is the duration of a particular call that has successfully accessed the trunked system and has been allocated a channel for a call. Service times are assumed to be exponential logarithmically with mean call duration H, where µ= is the mean service rate (average number of serviced calls per unit time). H is also called the average holding time. The probability that the service time of the nth serviced call will be less than some call duration s is given by

p{s_n<s} = 1 – e^-µsn s > 0 ----------------(10)

and the pdf of the service time is

pdf(s_n)=µ e^-µsn -------------------(11)

Where s_n is the service time of the nth call.

This trunking system represented by Erlang B formula is called an M/M/C/C queuing system (in waiting condition). The first M denotes a memoryless Poisson process for call arrivals, the second M denotes an exponential logarithmic service time for the users, the first C denotes the number trunked channels available, and the last V indicates a hard limit on the number of simultaneous users that are served.

The property of Markov chains can be used to derive the Erlang B formula.

Consider a discrete time stochastic process {X_n|n=0,1,2,3,…} that takes values from the set of nonnegative integers, so that the possible states of the process are i=0,1,... The process is said to be a Markov chain if its transition from the present state I to the next state I + 1 depends solely on the state I and not on previous states. Discrete time Markov chains enable the traffic to be observed at discrete observation points for specific traffic conditions. The operation of a practical trunked system is continuous in time, means in time domain, but may be analyzed in small time intervals δ, where δ is a small positive number. If N_k is the number of calls occupied channels in the system at time kδ, then N_k may be represented as

N_k = N(kδ) -------------------------(12)

Where N is a discrete random process representing the number of occupied channels at discrete times. N_k is a discrete time Markov chain with steady state occupancy probabilities which are identical to the continuous Markov chair, and can have values ranging on 0, 1, 2, … C.

The transition probability pdf_i,j describes the probability of channel occupies over a small observation interval, and is given by

pdf _i,j = p{N _k+1 = =i}------------------(13)

Using the equations (6) through (8), and letting δà0, we obtain

pdf₀₀ = 1-λδ+O(δ) -------------------(14)

pdf_ii = 1-λδ+µδ i ≥ 1 -------------------(15)

pdf_i,i+1 = λδ+O(δ) i ≥ 0 -------------------(16)

pdf_i,i-1 = µδ+O(δ) i ≥ 1 -------------------(17)

pdf_i,I = O(δ) j≠i, j≠i+1, j≠i-1 ------------(18)

The transition state diagram for the Markov chain is shown in Figure 1.

It is a Markov chain representation of a trunked system with C channels. To comprehend the Markov chain state diagram, assume that there are 0 channels being used by the system, i.e., there are no users. Over a small interval of time, the likelihood that the system will resume to use 0 channels is (1-λδ). The probability that there will be a change from 0 channels to 1 channel in use is given by λδ. On the other hand, if one channel is in use, the probability that the system will transit to 0 used channels is given by λδ. Similarly, the likelihood that the system will resume to use 1 channel is given by 1- λδ- λδ. All of the outgoing probabilities from a certain state sum to 1.

Over a long period of time, the system reaches steady state and has n channels use.

Figure 2 represents the steady state response of an LCC system.

Figure 2: A trunked LCC system at steady state with a number of channels in use.

At steady state, the probability of having n channels in use is equivalent to the probability of having (n-1) channels in use, times the transition probability λδ.

Hence, under steady state constraints,

λδpdf_n-1 = nµδpdf_n , n ≤ C----------(19)

equation (18) is known as the Global balance equation. Furthermore,

using the Global balance equation for different values of n, it is seen that

λδ pdf_n-1= pdf_n µδ, n=1,2,3,…,C --------------------(21)

λpdf_n-1 = pdf_n nµ---------------------------(22)

evaluating equation (21) for different values of n

and

subtracting equation (24) in equation (25) will yield the following equation

from equation (23), the probability of blocking for C trunked channels is

the total offered traffic is A = λH=λ/µ. Substituting this in to the equation (28), the probability of blocking is given by

finally we got the equation (29) which is the Erlang formula.

II. Erlang C model formula

The Erlang C formula is derived from the assumption that a queue is used to hold all requested calls which cannot be immediately assigned a channel. The Erlang C formula is given by

if no channels are immediately available, the call is delayed and held in a queue, and the probability that the delayed call is forced to wait for more than t seconds in the queue is given by

p(wait>t | call delayed) = ------------------ (31)

where c is the total number of available channels, t is the delay time of interest, and H is the average duration of a call. The probability that any caller is delayed in the queue for a wait time greater than t seconds is given by

p (wait>t) = p(call delayed) pr (wait>t | call delayed )-(32))

= pdf (call delayed)

The average delay D for all calls in a queued system is given by

D = dt ----------------(33)

D = pdf (call delayed) ---------------- (34)

Whereas the average delay of those calls which are queued is given by .

I. B. i. Derivation of Erlang C

Consider a system where c is the number of trunked channels. The assumptions used to derive the Erlang c formula are similar to those used to derive Erlang B, except for the additional stipulation that if an offered call cannot be assigned a channel, it is placed in a queue which has an infinite length. Each call is then serviced in the order of its arrival, the arrival process obeying a Poisson distribution given by

pdf{a(t+Δ) – a(t)=n} = -----------------(35)

where a is the number of call arrivals which are awaiting for service.

Just as the Erlang B model formula was derived using the assumption that all call inter-arrival, times are exponential (logarithmically) and independent the time between successive call arrivals is τ_n = t_n+1- t_n, and the distribution of the interarrival times is such that

pdf{ τ_n ≤ s} = 1-e^-λs, s ≥0 ----------------- (36)

As was the case for Erlang B, the service time for each user already on the trunked system is assumed to be exponential (logarithmically) and is given by

pdf{s_n ≤ s} = 1-e^-λs, s ≥0 ----------------- (37)

using discrete Markov chains with transition probabilities given in equations (13) to (18), the Erlang C formula is easily derived. Let pdf_i,j denote the transition probability of going from stet i to state j. the state. The state diagram for the system is shown in Figure 3.

The Erlang C formula is derived by assuming that the trunked system is a M/M/C/D queue, where c denotes the maximum number of simultaneous users and D is the maximum number of calls that may be held in the queue for service. If D is assumed to be infinite, then the system is an M/M/C/∞ queue, commonly referred to as simply M/M/C queue. If D is ∞, then P_k may denote the steady state probability of finding k calls in system (both served and queued calls). That is,

pdf_k = --------------------38)

where N_tis the total number of calls either using or waiting for the system at time t. in steady state, the probability that the system is in state k and makes a transition to state k-1 in the next transition interval is the same as the probability that the system is in state k-1 and makes a transition to state k. thus, from the state diagram shown in figure 3,

Figure 3: the transition probabilities represented as a Markov chain state diagram for Erlang C

--- (46)

the probability that an arriving call occurs when all C channels are busy and thus has to wait can be determined using the equation (43).

The above equation is valid for substituting for pdf₀ from the equation (46)

pdf(C channels are busy)=

But, A= substituting the above equation into equation (47), we obtain the following equation.

pdf(c channels are busy) = ---------(49)

above equation (49) is the expression for Erlang C model.

DISCUSSION:

The Erlang B model determines the probability that a call is blocked, and is a measure of the grade of service (GoS) for a trunked systems which provides no queuing for blocking calls which is an LCC systems, the Erlang B model formula is based on certain basic assumptions that are explained in the section I.

All requests are memory-less, i.e., may request a channel at any time. In Erlang C model formula, a queue is used to hold all reuested calls which cannot be assigned a channel immediately. The Erlang C formula derived by assuming the trunked system is a M/M/C/D queue; if Dis assumed to be finite, then system is an M/M/C/∞ queue commonly referred to as simply M/M/C queue. Equations/ expressions 29 and 49 are the Erlang model B and C formulas respectively.

CONCLUSIONS:

In this paper both Erlang model B and C formulas are derived mathematically. As mentioned in the discussion, expressions 29 and 49 are the Erlang model B and C formulas. These expressions can be gainfully utilized for the modeling of terrestrial propagation purposes for which simulation study can be conducted by using these expressions. Future research involves the simulation study using the Mat_Lab tool (a language used for the scientific research technical computing).

REFERENCES:

1. V. Rama Raju, “Solving Maxwell’s Equations Using Propagation Mechanisms - Parameters for Terrestrial Propagation Modeling”, Submitted to the IEEE Transactions on Wireless Communications, July, 2011.

2. V. Rama Raju, “Terrestrial Propagation Mechanisms for Terrestrial Propagation Modelling”, International Research Journal of Engineering and Technology, June, 2011 (Accepted).

3. V. Rama Raju,Tim Tozer, Dave Pierce, Niryong, “Terrestrial propagation Modeling: Propagation Models - Okumura, Cost 231, etc”, International Journal of Systemics, Cybernetics & Informatics, Pp: 7-17, October 2010

4. V. Rama Raju, “Terrestrial propagation Modeling: Identification of a suitable wireless technology”, International Journal of Systemics, Cybernetics & Informatics, Pp: 50-56, April 2010.

5. V. Rama Raju, Dave Pierce, Tim Tozer, V Malae, “Terrestrial propagation Modeling: Frequency Band Management 2.5 GHz to 60GHz ”, International Journal of Systemics, Cybernetics & Informatics, July 2010.

6. V. Rama Raju,Tim Tozer, Dave Pierce, Niryong, “Terrestrial propagation Modeling: Design, Planning and Implementation”, International Journal of Systemics, Cybernetics & Informatics, (Accepted).

7. Stallings W., “Local Networks”, McMillan Publishing Company, New York, 1990, reprint 2008.

8. Peyton Z. Peebles, Jr., “Probability, Random variables and random signal principles”, 18^th reprint, TATA-McGrawHill Publi., 2008 Edition.

9. Bertsekas, D., and Gallager R., “Data networks”, 2^nd Edition, Printice Hall Englewood Cliffs, NJ, 1992, (Reprint 2009)