Trunking Theory: Contd
1 Answer

7) A certain area is covered by a cellular radio system with 84 cells and a cluster size N. 300 voice channels are available for the system. Users are uniformly distributed over the area covered by the cellular system, and the offered traffic per user is 0.04 Erlang. Assume that blocked calls are cleared and the designated blocking probability is Pb =1 $\%$.

a. Determine the maximum carried traffic per cell if cluster size N =4 is used. Repeat for cluster sizes N=7 and 12.

b. Determine the maximum number of users that can be served by the system for a blocking probability of 1 $\%$ and cluster size N=4. Repeat for cluster sizes N=7 and N=12. Number of cells = 84

Number of available channels = 300

Traffic per user = 0.04 Erlang

GOS = 0.01

Number of Channels per cell $(N_c) =\frac{M(no \ of \ channels)}{N (cluster \ size)}$

For 4 ; $N_c$ =75

For 7 ; $N_c$ =42

For 2 ; $N_c$ =25

(a) Carried traffic per cell (from Erlang B chart)

For 4; traffic per cell =61 Erlang

For 7 ; traffic per cell =30.71 Erlang

For 12 ; traffic per cell=16.12 Erlang

(b) No of user = $\frac{A}{A_u}$

$A_u$=0.04 Erlang

For 4; GOS = 0.01,$N_c$ =75


For 7 ;U = 64491

For 12 ; U = 33852

8) The U.S. AMPS system is allocated 50 MHz of spectrum in the 800 MHz range and provides 832 channels. Forty-two of those channels are control channels. The forward channel frequency is exactly 45 MHz greater than the reverse channel frequency.

(a) Is the AMPS system simplex, half-duplex, or duplex? What is the bandwidth for each channel and how is it distributed between the base station and the subscriber?

(b) Assume a base station transmits control information on channel 352, operating at 880.560 MHz. What is the transmission frequency of a subscriber unit transmitting on channel 352?

(c) The A-side and B-side cellular carriers evenly split the AMPS channels. Find the number of voice channels and number of control channels for each carrier.

(d) Let’s suppose you are chief engineer of a cellular carrier using seven-cell reuse. Propose a channel assignment strategy for a uniform distribution of users throughout your cellular system. Specifically, assume that each cell has three control channels (120° sectoring is employed) and specify the number of voice channels you would assign to each control channel in your system.


Total bandwidth = 50MHz

Total number of channels provided( N) =832 channels

Therefore ,Bandwidth for each channel (BW)=$\frac{Bandwidth_{total}}{N} = \frac{50 \times 10^6}{832} = 60 kHz$

The AMPS System is duplex .The bandwidth of 60 kHz for this duplex channel is split into two a forward channel from base station to the subscriber and reverse channel from the subscriber to the base each with bandwidth of 30 KHz. The forward channel is exactly 45 MHz higher than the reverse channel.


Total number of control channel ($N_{CN}) = 42$

Total number of voice channel ($N_{VC}) = N - N_{CN} = 832 - 42= 790$

Number of voice channels for each carrier = $\frac{792}{2} = 395$ channels

Number of control channels for each carrier = $\frac{42}{2} = 21$

Number of cell sites=100

Traffic Intensity per cell(A) = 20

Bandwidth = 30kHz

Number of cells made by each user $\lambda$ 3 cells/hour

Average holding time per cell (H)=4 minutes = $\frac{4}{60}$ hours


Therefore, Traffic Intensity per user = ($Au)= \lambda \times H = \frac{3 \times 4}{60} =0.2$

Therefore number of users per cell = $\frac{A}{A_u} ={20}{0.2} = 100$ Users

9) A cellular system, has a total coverage area of 2000 sq. km and radius of each cell is 5 sq. km. Total number of radio channels available is 1000. Find the channel capacity for cluster size a) N=7 b) N=4 c) What inference do you draw about capacity and cluster size relation?

(a) Given,coverage area of cellular system = 2000 sq.km

Radius of each cell R =5 sq.km

No of radio channels available for handling traffic =1000

Area of a single cell = $\frac{\text{Coverage area}}{\text{area of one cell}} = \frac{2000}{64.9525} = 30.79$ cells



For cluster size (N) = 7

Replication rate , (M) = $\frac{30.79}{7} = 4$

Channel Capacity (C) =KNM = $\frac{1000}{7} \times 7 \times 4= 4000$ channels.

(b) For cluster size (N) = 4

Replication rate , (M) = $\frac{30.79}{4} = 8$

Channel Capacity (C) =KNM = $\frac{1000}{4} \times 4 \times 8= 8000$ channels.

(c) Yes the channel capacity increases as the cluster size decreases because it reduces co channel and frequency range of channels

Try this Out:

1) Consider a cellular system consisting of 5 users. Each user makes an average of 3 calls per hour. Each call, on average, lasts for 4 minutes. What is the total offered traffic (in Erlang) ?

Ans: Load per user $(A_u)=\lambda \times H =3 calls/(60 \times 60) \times \frac{4}{60} = 0.2$ Er

Total load A = Number of users $\times A_u$= 5 x 0.2 Er=1 Erlang

2) What is the maximum system capacity per channel in Erlangs when providing a blocking probability of 2 $\%$ with C = 40 channels

(Use the Erlang-B calculator given in http://www.erlang.com/calculator/erlb/ )

Ans: From Erlang-B Calculator.

For 2 $\%$ blocking probability and 40 channels

Total Capacity = 30.95 Erl

Therefore, Capacity per Channel = $\frac{30.95}{40} = 0.77$ Erl

3) A cellular system has a cluster size N. It is given that a total of 300 voice channels are available, and users are uniformly distributed over the geographic area covered by the cells. Each user generates 0.04 Erlangs of traffic. Assuming that blocked calls are cleared and that the target blocking probability is 1 $\%$ .

(Use the Erlang-B calculator given in http://www.erlang.com/calculator/erlb/ )

What is the maximum number of users per cell that can be supported for cluster size N = 4 ?

For N=4,No of Voice Channels per cell = $\frac{300}{64} =75$ channels.

From Erlang-b calculator ,for 75 channels and 1 $\%$ blocking probability, total traffic = 60.7 Erl.

Traffic per user =0.04 Erl

Therefore, No of User per cell = $\frac{60.7}{0.04} =1517.5=1517$ Users

4) For the system in previous question, what should be the reduction in offered traffic to maintain the same Grade-of-Service (GOS) if one channel in each cell is reserved for handoff traffic. (Assume cluster size is N = 4 )

$\quad$ The offered traffic per cell previously = 60.7 Erl

$\quad$ New number of channels in each cell = 75-1=74 channels.

$\quad$ Using Erlang-b Calculator ,new offered traffic per cell =59.8 Erl.

$\quad$ Therefore,Reduction in offered traffic = 60.7-59.8 = 0.8 Erl.

5) For a N =7 system with blocking P = 1 $\%$ and average call length of two minutes, find the capacity loss (in terms of No. of users per cell) due to trunking for 60 channels / cell when going from omni-directional antenna to 60 degree sectored antennas. Assume blocked calls are cleared and the average user call rate is 1 call per hour.

N=9{Hexagonal Geometry}

$P_r(blocking)$= 1 $\%$ =0.01.

$\lambda$=1 call/hour.

$H= \frac{2}{60}$ hours

$A_n = H = \frac{2}{60} \times 1 = \frac{1}{30}$Erl.

Omni–directional Antenna:-

$60^0$ Channels/cell

From Erlang b calculator ,traffic offered / cell = 46.9 Erl.

Number of users/cell=$\frac{46.9}{1/30} = 1407$ users/cell.

60 sectorization :-

10 Channels / cell sector.

From Erlang B Calculator ,traffic offered per cell per sector =4.45.

No of user/cell/sector =$\frac{4.45}{1/30} = 133$

No of Users/cell = $133\times 6=798$ users/cell.

Therefore, Reductions is capacity = 1407-998=609 users/call

Ans: 609 users/cell

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