The Cellular ConceptSystem Design Fundamentals

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Table of ContentsFrequency ReuseChannel Assignment StrategiesHandoff Strategies

Prioritizing HandoffsPractical Handoff Considerations

Interference and System CapacityPower ControlTrunking and Grade of ServiceImproving Coverage and Capacity in Cellular SystemsTrunking Theory

Frequency Reuse

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Cellular System Design Considerations

To solve problems of spectral congestion and user capacity.Replacing a single, high power transmitter with many low power transmitters.Neighboring base stations are assigned different groups of channels so that the interference between base stations is minimized.Available Channels are distributed throughput the geographic region and may be reused as many times as necessary.With fixed number of channels to support an arbitrarily large number of subscribers.

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Concepts of Frequency Reuse

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Cellular Networks and Frequency Reuse

One important characteristic of cellular networks is the reuse of frequencies in different cells.By reuse frequencies, a high capacity can be achieved.However, the reuse distance has to be high enough, so that the interference caused by subscribers using the same frequency (or an adjacent frequency) in another cells is sufficiently low.To guarantee an appropriate speech quality, the carrier-to-interference-power-ratio (CIR) has to exceed a certain threshold CIRmin which is 9 dB for the GSM system.

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HexagonsHexagonal cell shape has been universally adopted.The actual radio coverage of a cell is known as the footprint and is determined from field measurements or propagation prediction models.Base stations can be placed at:

The cell center – center-excited cells – omni-directional antennas.The cell vertices – edge-excited cells – sectored directional antennas.

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Analysis Model of Channel ReuseS: # of duplex channels available for usek: # of channels assigned for a cellN:# of cells sharing S channels – Cluster Size

C: Total # of duplex channels with freq. reuseM: # of Replica of cluster

kNS =

MSMkNC ==

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Reuse Factor

Reuse Factor = i2+ij+j2; i,j are non-negative integers.

Due to the fact that the hexagonal geometry has exactly six equidistant neighbors and that the lines joining the centers of any cell and each of its neighbors are separated by multiples of 60 degrees, there are only certain cluster sizes and cell layouts which are possible.

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Co-channel Neighbor Cells

Move i cells along any chain of hexagons.Turn 60o counter-clockwise and move j cells.For example:

N=19: i =3, j =2;N=12: i =2, j =2;N=7; i =2, j =1;

Channel Assignment Strategies

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Channel Assignment StrategiesFixed Channel Allocation

Dynamic Channel Allocation

Hybrid Channel Allocation

Borrowed Channel Allocation

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Fixed Channel AssignmentEach cell is allocated a predetermined set of voice channels.Any call attempt within the cell can only be served by the unused channels in that particular cell.Probability of blocking is high.

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Dynamic Channel Assignment Strategy

Channels are not allocated to different cells permanently.Each time a call request is made, the serving base station requests a channel from the MSC.The MSC allocates a channel to the requested cell following an algorithm that takes into account the likelihood of future blocking within the cell, the frequency of use of the candidate channel, the reuse distance of the channel, and other cost functions.MSC only allocates a given frequency if that frequency is not presently in use in the cell or any other cell which falls within the minimum restricted distance of frequency reuse to avoid co-channel interference.

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Dynamic Channel Assignment Strategy

MSC has to collect real-time data on channel occupancy, traffic distribution, and radio signal strength indications (RSSI) of all channels on a continuous basis.Reduce the likelihood of blocking at the expense of increasing the storage and computational load.

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Borrowing Strategy

Modified from fixed channel assignment strategies.

A cell is allowed to borrow channels from a neighboring cell if all of its own channels are already occupied.

The MSC supervises such borrowing procedures and ensures that the borrowing of a channel does not disrupt or interfere with any of the calls in progress in the donor cell.

Handoff Strategies

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Handoff / Handover

In a cellular network, the process to transfer the ownership of a MS from a BS to another BS.Handoff not only involves identifying a new BS, but also requires that the notice and control signals be allocated to channels associated with the new base station.Usually, priority of handoff requests is higher than call initiation requests when allocating unused channels.Handoffs must be performed successfully and as infrequently as possible and be imperceptible to the uses.

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Handoff / HandoverHandover Occasions

Bad signal quality on current channelnoise or interference

Traffic overload in current cellload balancing

Handover Indicator: The parameters to monitor to determine HO occasion

RSSI, in ensemble average sense.Bit Error Rate (BER)/Packet Error Rate (PER), more accurate.

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Handoff / Handover

Need to specify an optimum signal level to initiate a handoff.Minimum useable signal for acceptable voice quality at the base station receiver is normally taken as between -90 dBm to -100 dBm.

If Δis too large, unnecessary handoffs may occur.If Δis too small, there may be insufficient time to complete a handoff.

useableminimumhandoff PrPr −−=∆

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Illustration of a handoff scenario at cell boundary

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Illustration of a Handoff Scenario at Cell Boundary

Figure (a) demonstrates the case where a handoff is not made and the signal drops below the minimum acceptable level to keep the channel active.The dropped call event in figure (a) can happen when there is an excessive delay by the MSC in assigning a handoff or when the threshold Δ is set too small for the handoff time the system.Excessive delays may occur during high traffic conditions due to computational loading at the MSC or due to the fact that no channels are available on any of the nearby base stations.

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Handoff / HandoverDuring handoff, it is important to ensure that the drop in the measured signal level is not due to momentary fading and that the mobile is actually moving away from the serving base station.The base station monitors the signal level for a certain period of time before a hand-off is initiated.The time over which a call may be maintained within a cell, without hand-off, is called the dwell time.

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Handoff in 1st Generation Cellular Systems

Signal strength measurements are made by the base stations and supervised by the MSC.Base station monitor the relative location of each user.Locator receiver is used to determine signal strengths of users in neighboring cells and is controlled by the MSC.Based on the information from locator receiver, MSC decides if a handoff is necessary or not.

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Handoff in 2nd Generation TDMA Systems

Handoff decisions are mobile assisted.In mobile assisted handoff (MAHO), every mobile measures the received power from surrounding base stations and reports the results to the serving base station.MAHO enables the call to be handed over at a much faster rate.

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Handoff or HandoverMobile Assistant Handover – more efficient.

GSM: MS monitors all BSsMS reports the measurements to the BSMSC makes decision

USDC (IS-54/136): BSs monitor all MSs. When a MS is leaving the cell, the BS sends it a measurement orderThe MS begins its measurement and reportsMSC makes the Handover decision.

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Handover Algorithms (IS-95 vs. WCDMA)

Basic IS-95 handover algorithm uses absolute threshold algorithm.

WCDMA handover algorithm users relative threshold algorithm.

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Absolute Threshold Handover

Th_Add

Th_Drop

Time

Eb/No

NeighborSet

CandidateSet

ActiveSet

NeighborSet

(1) (2) (3) (4) (5) (6)

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Basic IS-95 HO Algorithm1. Pilot strength exceed T_Add. MS sends a Pilot Strength

Measurement Message and transfers pilot to the Candidate Set.2. BS sends a Handover Direction Message.3. Mobile station transfers pilot to the Active Set and sends a

Hanover Completion Message.4. Pilot strength drops below T_Drop. MS starts the handover drop

timer.5. Handover drop timer expires. MS sends a Pilot Strength

Measurement Message.6. BS sends a Handover Direction Message. MS moves pilot from

the Active Set to the Neighbor Set and sends a Handover Completion Message.

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Problems with Absolute Threshold Algorithm

Some locations in the cell receive only weak pilots (requiring a lower handover threshold).

Some locations in the cell receive a few strong an dominant pilots (requiring a higher handover threshold).

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Relative Threshold HO

Strongest Pilot in Active SetEc/Io

Window_Add

Window_Drop

T_DropT_AddTime

MS_Ec/Io

MS AS MS

AS_Th AS_Th_Hyst

AS_Th_Hyst

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Active vs. Monitored SetActive Set (AS): User information is sent from all these cells and they are simultaneously demodulated and coherently combined.

Monitored Set (MS): Cells, which are not included in the active set, but are monitored according to a neighboring list assigned by the UTRAN.

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Soft Handover Algorithm(for Active Set limit = 2)

AS_Th -AS_Th_HystAS_Rep_Hyst

AS_Th + AS_Th_Hyst

Cell 1 Connected

CPICH 1

CPICH 2

CPICH 3

Time

Ec/No

∆T ∆T ∆T

Event AAdd Cell 2

Event BReplace Cell 1 with Cell 3

Event CRemove Cell 3

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Intersystem HandoffIntersystem handoff happens when a mobile moves from one cellular system to a different cellular system.The MSCs involved in the two cellular systems are different.Compatibility between the two MSCs must be determined.

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Prioritizing HandoverGuard Channel Concept : Use reserved guard channel for handover.

Disadvantage: Reducing the total carrier traffic.

Queuing of Handover Requests: To prevent forced termination by queuing the request.

Queuing of handoffs is possible due to the fact that there is a finite time interval between the time the received signal level drops below the handoff threshold and the time the call is terminated due to insufficient signal level.

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Practical Handover ConsiderationProblem 1: Simultaneous traffic of high speed and low speed mobiles.

Small cell → high speed mobile → frequent handoff large cell → Reduce capacity

Solution: Umbrella Cell - Cell Split or Hierarchical Cell Structure

By using different antenna heights and different power levels, it is possible to provide large and small cells which are co-located at a single location.Small cell for low speed mobileLarge cell for high speed mobileNeed Strength Detection and Handoff control.

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The Umbrella Cell Approach

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Practical Handover ConsiderationProblem 2: Cell Dragging

Caused by pedestrian users that provide a very strong signal to the base station.Often occurs in an urban environment when there is a line-of-sight (LOS) radio path between the subscriber and the base station.As the user travels away from the base station at a very low speed, the average signal strength does not decay rapidly and the received signal at the BS may be above the handoff threshold, thus a handoff may not be made.Creates a potential interference and traffic management problem.

Solution: Careful arrangement of handoff threshold and radio coverage parameters.

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Handoff Control Parameters

Δ≅ 6~12 dB (AMPS) → Δ≅0~6 dB (GSM)

Typical time to make a handoff – once the signal level is blow the handoff threshold : 10 sec (AMPS) → 1~2 sec (GSM)

The faster handoff process supports a much greater range of options for handling high speed and low speed users and provides the MSC with substantial time to rescue a call that is in need of handoff.

useableminimumhandoff PrPr −−=∆

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Handoff MiscellaneousIntra-frequency Handoff: handoffs in the same system and carrier.Inter-frequency Handoff: handoffs between same systems and different carriers.

May be used for handoff between different cell layers of the multi-layered cellular network, when the cell layers use different carrier frequencies.

Inter-system Handoffs: handoffs between different systems.Inter-frequency and inter-system handoffs may be used for coverage or load balancing reasons.

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Frequency Utilization with WCDMA

Frequency

Power

5.0-5.4 MHz 5.0-5.4 MHz

4.2-5.0 MHz 4.2-5.0 MHz

Operator band 15 MHz

3 cell layers

Another UMTSoperator

Another UMTSoperator

Uplink: 1920-1980 MHzDownlink: 2110-2170 MHz

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Intra-Frequency Handoff

Hard Handoff: assign different radio channels during a handoff.

Soft Handoff: the ability to select between the instantaneous received signals from a variety of base stations.

Soft handoff exploits macroscopic space diversity provided by the different physical locations of the base stations.

Softer Handover: A mobile station is in the overlapping cell coverage area of two adjacent sectors of a base station.

Interference and System Capacity

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Interference

The major source limiting cellular system capacity comes from interferences (as oppose to noise).Interference has been recognized as a major bottleneck in increasing capacity and is often responsible for dropped calls.

Major Types of Interference:Co-Channel InterferenceAdjacent Channel Interference

Intra-Cell TypeInter-Cell Type

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Co-channel Cells and Interference

In a given coverage area, there are several cells that use the same set of frequencies. These cells are called co-channel cells.The interference between signals from co-channel cells is called co-channel interference.Unlike thermal noise which can be overcome by increasing the signal-to-noise ratio (SNR), co-channel interference can’t be overcome by simply increasing the carrier power because an increase in carrier power increases the interference to neighboring co-channel cells.To reduce co-channel interference, co-channel cells must be physically separated by a minimum distance.

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Hexagon

Rr ⋅=23

R

r

2 2( ) 3 3Cell Area 6 2 32 2

r R R r⋅= ⋅ = ⋅ =

Rr ⋅=23

R

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Hexagon

A cell centered at (u,v)

vuvuy

uux

rjrivu

o

o

+=+⋅=

⋅=⋅=

=

2)30sin(

23)30cos(

)2,2(),(

i=3

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Distance between (ua, va) and (ub, vb)

For special case (distance from origin)

( ) ( ) ( ) ( )

2 2 2

2 2

2 2

2 2 2

( ) ( )

3 3 2 2 2 2

(2 ) ( ) ( ) ( )( )

ab a b a b

a ba b a b

a b a b a b a b

a b a b a b a b

D x x y y

u uu u v v

u u v v u u v v

r i i j j i i j j

= − + −

⎛ ⎞ ⎛ ⎞= − + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

= − + − + − ⋅ −

⎡ ⎤= ⋅ − + − + − −⎣ ⎦

2 22D r i j ij= ⋅ + +

Hexagon Distance

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Hexagonal Cluster

Hexagonal ClusterEach cluster is surrounded by six similar clusters with the same orientationEach cluster has a total area equivalent to what can be called a “super-hexagon”

view a cluster as a “hexagon”

2 22D r i j ij= ⋅ + +

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Super-Hexagon Concept

D=Frequency Reuse Distance

( )( )

2

2

2 2

'2

2 3 '

2 3

Cluster Size

Super Hexagon

Cell

DR

AN

A

R

r

i j i j

−

=

=

=

= + + ⋅=

D

R’

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Co-Channel Reuse Ratio

The signal to co-channel interference ratio is independent of the transmitted power and becomes a function of the radius of the cell (R) and the distance between centers of the nearest co-channel cells (D). Co-channel Reuse Ratio Q : The spatial separation between co-channel cells relative to the coverage distance of a cell.

NR

Nr

jijirD

3

2

2 22

⋅=

⋅⋅=

⋅++⋅⋅=

NRDQ 3==

Rr ⋅=23

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CO-Channel Reuse Ratio

Small Q Small N Large Capacity

Large Q Large N Small Level of Co-Channel Interference

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Co-Channel Signal to Interference Ratio (SIR)

SIR: Signal to Interference Ratio.Consider only first tier interference.Assuming all interfering BS’s are equal-distance.

∑=

=0

1

i

iiI

SIS

NR 3⋅

00 0

1

( 3 )Worst Case SIR=n n n

ti n

nt i

i

PR R Ni D iPd

− −

−−

=

≈ =

∑Example: i0=6, n=4N=3 ⇒ SIR=11.3 dBN=7 ⇒ SIR=18.7 dB

n

r ddPP

−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

00

(Mobile)

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7-Cell Co-channel Signal to Interference Ratio

An approximation of the exact geometry.

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7-Cell Co-Channel Signal to Interference Ratio – An Approximation

2( ) 2( ) 2

n

n n n

S RI D R D R D

−

− − −=− + + +

( ) ( )4 4 4

12 1 2 1 2

SI Q Q Q− − −=

− + + +

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Adjacent Channel InterferenceAdjacent channel interference results from imperfect receiver filters which allow nearby frequencies to leak into the passband.Adjacent channel interference can be minimized through careful filtering and channel assignments.Channels are allocated such that the frequency separation between channels in a given cell is maximized.

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Near-Far Effect

A nearby transmitter (which may or may not be of the same type as that used by the cellular system) captures the receiver of the subscriber.Alternatively, the near-far effect occurs when a mobile close to a base station transmits on a channel close to one being used by a weak mobile. The base station may have difficulty in discriminating the desired mobile user from the close adjacent channel mobile.

Power Control

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Power Control for 2G Cellular Systems

Power levels transmitted by subscriber unit are under control by the serving base stations.Power control is to ensure that each mobile transmits the smallest power necessary to maintain a good quality link on the reverse channel.Power control not only helps prolong battery life for the subscriber unit, but also dramatically reduces the reverse channel S/I in the system.

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Power Control in 3G (WCDMA)Tight and fast power control is perhaps the most important aspect in WCDMA in particular on the uplink. Without it, a single overpowered mobile could block a whole cell.Near-Far problem of CDMA: A MS close to the base station may be overpowered and block a large part of the cell.Power control in WCDMA:

Open-loop power controlClose-loop power control

Inner-loop power controlOuter-loop power control

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Open Loop Power Control in WCDMAAttempt to make a rough estimation of path loss by measuring downlink beacon signal.Disadvantage: Far too inaccurate, because fast fading is essentially uncorrelated between uplink and downlink, due to the large frequency separation of uplink and downlink band of the WCDMA FDD mode.Open-loop power control is used in WCDMA to provide a coarse initial power setting of the MS at the beginning of a connection.

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Outer Loop Power Control in WCDMA

To adjusts the target SIR setpoint in the BS according to the individual radio link quality requirements, usually defined as BER or FER.The required SIR for FER depends on the mobile speed, multipath profile, and data rate.Should the transmission quality is decreasing, the Radio Network Controller (RNC) will command the Node B to increase the target SIR.Outer loop power control is implemented in RNC because there might be soft handover combining.

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Base station performs frequent estimates of the received Signal-to-Interference Ratio (SIR) and compares it to a target SIR.

If the measured SIR is higher than the target SIR, the base station will command the MS to lower the power.If SIR is too low, it will command the MS to increase its power.

The power control is operated at a rate of 1500 times per second.

Inner-loop Power Control in WCDMA Uplink

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Inner-loop Power Control in WCDMA Downlink

Adopt same techniques as those used in uplink.Operate at a rate of 1500 times per second.

There is no near-far problem in downlink. Purposes for downlink closed-loop power are:

Provide a marginal amount of additional power to MS at the cell edge as they suffer from increased other-cell interference.Enhancing weak signals caused by Rayleigh fading at low speeds when other error-correcting methods (interleaving and error correcting codes) doesn’t work effectively.

Trunking and Grade of Service

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Trunking System

Trunking system: A mechanism to allow many user to share fewer number of channels.

Not every user calls at the same time.

Penalty: Blocking Effect.If traffic is too heavy, call is blocked!!Small blocking probability is desired.

There is a trade-off between the number of available circuits and blocking probability.

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Grade of Service (GOS)Erlang: The amount of traffic intensity carried by a channel that is completely occupied.For example, a radio channel that is occupied for 30 minutes during an hour carries 0.5 Erlangs of traffic.Grade of Service (GOS) is a measure of the ability of a user to access a trunked system during the busiest hour.GOS is typically given as the likelihood that a call is blocked, or the likelihood of a call experiencing a delay greater than a certain queuing time.

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Definitions of Terms Used in TrunkingTheory

Set-up Time: The time required to allocate a radio channel to a user.Blocked Call: Call which can’t be completed at time of request, due to congestion.Holding Time: Average duration of a typical call. Denoted by H (in second).Traffic Intensity: Measure of channel time utilization, which is the average channel occupancy measured in Erlangs. This is a dimensionless quantity and may be used to measure the time utilization of single or multiple channels. Denoted by A.

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Definitions of Terms Used in TrunkingTheory

Load: Traffic intensity across the entire trunked radio system, measured in Erlangs.Grade of Service (GOS): A measure of congestion which is specified as the probability of a call being blocked (Erlang B), or the probability of a call being delayed beyond a certain amount of time (Erlang C).Request Rate: The average number of call requests per unit time. Denoted by μ second-1.

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Computation of GOS

Au=λHAu : each user generate a traffic intensity of Au Erlangλ : average number of call request per unit time for each user.H : average duration of a call.

A=UAuA: Total offered traffic intensity.U: Total users in a system.

Ac=UAu/CC : # of channels in a truncking systemAc: channel intensity per channel

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Computation of GOSThere are two types of trunked systems:

No queueing for call requests:for every user who requests service, it is assumed there is no setup time and the user is given immediate access to a channel if one is available.If no channels are available, the requesting user is blocked without access and is free to try again later.Called blocked calls cleared.GOS: Erlang B formula determines the probability that a call is blocked.

A queue is provided to hold calls which are blocked.If a channel is not available immediately, the call request may be delayed until a channel becomes available.Called blocked calls delayed.GOS: Erlang C formula gives the likelihood that a call is initially denied access to the system.

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Trunked Systems with no Queuing for Call Requests – blocked calls cleared.

Blocking Probability -- Erlang B Formula:

There are infinite number of users.Call requests are memoryless; both new and blocked users may request a channel at any time.Service time of a user is exponentially distributedTraffic requests are described by Poisson model.Inter-arrival times of call requests are independent and exponentially distributed.

GOS

kA

CA

blocking C

k

k

C

==

∑=0 !

!]Pr[AMPS cellular system is designedfor a GOS of 2% blocking.

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Trunked Systems with Queuing for Call Requests – Blocked Calls Delayed

Erlang C Formula – the likelihood of a call is initially denied to access a channel:

∑−

=

⎟⎠⎞

⎜⎝⎛ −+

=> 1

0 !1!

]0Pr[ C

k

kC

C

kA

CACA

Adelay

))(exp(]0Pr[ ]0|Pr[]0Pr[]Pr[

HtACdelaydelaytdelaydelaytdelay

−−>=>>>=>

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Trunked Systems with Queuing for Call Requests – Blocked Calls Delayed

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

The average delay for those calls which are queued is given by:

ACHdelayD−

>= ]0Pr[

ACH−

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Erlang B Chart

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Erlang C Chart

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Trunking EfficiencyTrunking Efficiency is a measure of the number of users which can be offered a particular GOS with a particular configuration of fixed channels.The way in which channels are grouped can substantially alter the number of users handled by a trunked system.

From Table 3.4, for GOS=0.0110 trunked channels can support 4.46 Erlangs.Two 5 trunked channels can support 2x1.36=2.72 Erlang.10 trunked channels support 64% more traffic than two 5 channel trunks do.

Improving Capacity in Cellular Systems

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System Expansion TechniquesAdding New ChannelsFrequency borrowingCell SplittingSectoring / SectorizationChange of Cell PatternCoverage zone

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Cell SplittingCell splitting is the process of subdividing a congested cell into smaller cells, each with its own base station and a corresponding reduction in antenna height and transmitter power.

Cell splitting increases the capacity of a cellular system since it increases the number of times that channels are reused.

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Cell SplittingCell splitting⇒ small cells (microcells)

Same service area

More cells in the service area, more capacity.

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Cell Splitting

Cell Splitting(Hot Spot)

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Transmit Power for Split cellThe transmit power of the split cell must be reduced.

For example, if new cell radius is half of that of old cell and the path loss exponent n = 4:

ntr RPP −∝ 1boundary] cell oldat [

ntr RPP −∝ )2(boundary] cell newat [ 2

boundary] cell oldat [boundary] cell newat [ rr PP =

161

2t

tPP =

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SectoringThe technique for decreasing co-channel interference and thus increasing system capacity by using directional antennas is called sectoring.

The factor by which the co-channel interference is reduced depends on the amount of sectoring used.

# of antenna ↑ , # of handover ↑ , trunking efficiency ↓

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Sectoring

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Sectoring

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Chang of cell Pattern

N=7, ⇒ CT/7 channels per cell N=3, ⇒ CT/3 channels per cell

high SIR low SIR (⇐ by-product)

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A Novel Microcell Zone Concept

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A Novel Microcell Zone ConceptAny channel may be assigned to any zone.When a mobile travels from one zone to another, there is no handover.A given channel is active only in a particular zone. Thus interference is reduced and capacity is increased.Without the degradation in trunking efficiency.

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Summaries of Improving Capacity in Cellular Systems

While cell splitting increases the number of base stations in order to increase capacity, sectoring and zone microcells rely on base station antenna placements to improve capacity by reducing co-channel interference.Cell splitting and zone microcell techniques do not suffer the trunking inefficiencies experienced by sectored cells, and enable the base station to oversee all handoff chores related to the microcells, thus reducing the computational load at the MSC.

Trunking Theory

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IntroductionThere are two major classes of trunked radio systems:

Lost Call Cleared (LCC)Lost Call Delayed (LCD)

Lost Call Cleared systemQueueing is not provided for call requests.When a user requests service, the user is given immediate access to a channel if one is available.If no new channels are available, the call is blocked without access to the system. The user is free to try the call again later.Calls are assumed to arrive with a Poisson distribution.The time between successive calls is exponentially distributed.The Erlang B formula describes the grade of service (GOS) as the probability that an user will experience a blocked call.

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IntroductionAssumed that there are infinite number of users.Assume that all blocked calls are instantly returned to an infinite user pool, and may be retried at any time in the future.

Lost Call Delayed system:Queues are used to hold call requests that are initially blocked.When a channel is not immediately available, the call request may be delayed until a channel becomes available.The Erlang C formula gives the likelihood that all channels are in use. It is also necessary to know the probability of how long the cal must be delayed before a channel is available.The GOS is measured by the probability that calls will have delayed greater than t seconds.Assume that all calls in the queue are eventually serviced.

Erlang B Formula

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Basic AssumptionsCall requests are memoryless, implying that all users, including blocked users, may request a channel at any time.The probability of a user occupying a channel (call the service time) is exponentially distributed.Traffic requests are described by a Poisson distributionwhich implies exponentially distributed call inter-arrival times.Inter-arrival times of call requests are independent of each other.There infinite number of users and there are finite number of channels available in the trunked pool.

WITS Lab, NSYSU.96

Derivation of Erlang BConsider 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).Let H be the average call holding time (average call duration).If A is the offered load for the trunked system, A=λH.The probability that a call requested by a user will be blocked is given by:

Pr[Blocking]=Pr[None of the C channels are free]Calls arrive according to the Poisson distribution:

( ) ( ){ } ( )Pr for = 0, 1, 2...!

nea t a t n nn

λτ

τ λτ−

+ − = =

WITS Lab, NSYSU.97

Derivation of Erlang Ba(t) is the number of call requests (arrivals) that have occurred since t = 0.The Poisson process implies that the time of the nth call arrival and the interarrival times between successive call requests are mutually independent.The interarrival times between call requests are exponentially distributed and mutually independent, and the probability that the interarrival time will be less than some time s is given by Pr(τn≤s)=1-e-λs, s≥0 where τn is the interarrival time of the nth arrival and τn=tn+1-tn, where tn is the time at which the nth call request arrived.The probability density function forτn is:

( ) , 0.nn np e λττ λ τ−= ≥

WITS Lab, NSYSU.98

Derivation of Erlang BFor every t ≥0 andδ≥0

where O(δ) is the probability of more than one call request arriving over the time interval δ and is a function ofδ such that

The probability of n arrivals in δseconds is given by:

( ) ( ){ } ( )( ) ( ){ } ( )( ) ( ){ } ( )

Pr 0 1

Pr 1

Pr 2

a t a t O

a t a t O

a t a t O

δ λδ δ

δ λδ δ

δ δ

+ − = = − +

+ − = = +

+ − ≥ =

( )0

lim 0O

δ

δδ→

⎧ ⎫=⎨ ⎬

⎩ ⎭

( ) ( ){ } ( )Pr!

nea t a t nn

λδ

δ λδ−

+ − = =

(A)

WITS Lab, NSYSU.99

Derivation of Erlang BThe user service time is the duration of a particular call that has successfully accessed the trunked system.Service times are assumed to be exponentially distributed with mean call duration H, where μ=1/H is the mean service rate.The probability density function of the service time is

where sn is the service time of the nth call.This trunking system is called an M/M/C/C queueing system. The first M denotes a memoryless Poisson process for call arrivals, the 2nd M denotes an exponentially distributed service time, the first C denotes the number of channels available, and the last Cindicates a hard limit on the number of simultaneous users that are served.

( ) nsnp s e µµ −=

WITS Lab, NSYSU.100

Derivation of Erlang BConsider a discrete time stochastic process {Xn|n=0, 1, 2,…} that takes values from the set of nonnegative integers, so that the possible states of the process are i=0,1, 2, …. The process is said to be a Markov chain if its transition from the present state ito the next i+1 depends solely on the state i and not on previous states.At time kδ, the number of calls (occupied channels) Nk in the system may be represented as Nk = N(kδ), where N is a discrete random process representing the number of occupied channels.The transition probability Pi,j, which describes the probability of channel occupancies over a small observation interval, is given by

{ }, 1Pr |i j k kP N j N i+= = =

WITS Lab, NSYSU.101

Derivation of Erlang BUsing Equation (A) and letting δ→0, we obtain:

( )( )

( )( )

( )

, 1

, 1

,

1

1 1

0

1

, 1, 1

oo

ii

i i

i i

i j

P O

P O i

P O i

P O i

P O j i j i j i

λδ δ

λδ µδ δ

λδ δ

µδ δ

δ

+

−

= − +

= − − + ≥

= + ≥

= + ≥

= ≠ ≠ + ≠ −

WITS Lab, NSYSU.102

Derivation of Erlang BAt steady state, we have the Global Balance Equation:

As a result, we have:1 , n nP n P n Cλδ µδ− = ≤

01

C

nn

P=

=∑1

20

1 2 1 0

0 0 01 1

0

0

1 1 2 2

1 1 ! 1 1! !

11!

n n

n in C C

n n ii i

nC

n

P P n

PP P P P

P P P P n P Pn i

P

n

λ µ

λ λ λµ µ µ

λ µ λµ λ µ

λµ

−

= =

=

=

⎛ ⎞= = = ⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞⎛ ⎞= ⇒ = = − = −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

=⎛ ⎞⎜ ⎟⎝ ⎠

∑ ∑

∑

WITS Lab, NSYSU.103

Derivation of Erlang BThe probability of blocking for C trunked channels is:

The total offered traffic is A=λH=λ/μ. The Erlang B formulawhich gives the probability of blocking can be obtained by:

0

0

1!1

! 1!

C

C

c nC

n

CP P

Cn

λµλ

µ λµ=

⎛ ⎞⎜ ⎟⎛ ⎞ ⎝ ⎠= =⎜ ⎟

⎝ ⎠ ⎛ ⎞⎜ ⎟⎝ ⎠

∑

0

1!1!

C

c Cn

n

ACP

An=

=

∑

Erlang C Formula

WITS Lab, NSYSU.105

Derivation of Erlang CAssumption: if an offered call can’t be assigned a channel, it is placed in a queue which has an infinite length.The Erlang C formula is derived by assuming that the trunkedsystem 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.

WITS Lab, NSYSU.106

Derivation of Erlang CFrom the state diagram:

1 1

1 1

0

0

1 for for

1 for for

1 !

1 1

!

k k k k

k k k k

k

k k

k C

P k P k C P P k Ck

P C P k C P P k CC

P k Ck

P

P k CC C

λλδ µδµ

λλδ µδµ

λµ

λµ

− −

− −

−

⎛ ⎞= ≤ ⇒ = ≤⎜ ⎟

⎝ ⎠⎛ ⎞

= ≥ ⇒ = ≥⎜ ⎟⎝ ⎠

⎧⎛ ⎞≤⎪⎜ ⎟

⎪⎝ ⎠⇒ = ⎨⎛ ⎞⎪

≥⎜ ⎟⎪⎝ ⎠⎩

WITS Lab, NSYSU.107

Derivation of Erlang C

( )

0

1

0 1

1

01

0 1

1

Since =1, we have:

1 11 ...... .... 1!

1 1 1 1 1! !

1 1 1 1! ! 1

kk

C

C C

k kC

k Ck k c

k CC

k

P

PC C

Pk C C

P

k kC

λ λµ µ

λ λµ µ

λ λµ µ λ

µ

∞

=

+

+ −

− ∞

−= =

−

=

⎡ ⎤⎛ ⎞ ⎛ ⎞+ + + + =⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞ ⎛ ⎞

⇒ + + =⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

⇒ =⎛ ⎞ ⎛ ⎞

+⎜ ⎟ ⎜ ⎟ ⎛ ⎞⎝ ⎠ ⎝ ⎠ −⎜ ⎟⎝ ⎠

∑

∑ ∑

∑

WITS Lab, NSYSU.108

Derivation of Erlang C

[ ]

0

0

0

channels are busy

1 1!

1 1!

1 1 1! 1

r kk C

k

k Ck C

C k C

k Ck C

C

P C P

PC C

PC C

PC C

C

λµ

λ λµ µ

λ λµ µλ

µ

∞

=

∞

−=

−∞

−=

=

⎛ ⎞= ⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞= <⎜ ⎟ ⎛ ⎞⎝ ⎠ −⎜ ⎟

⎝ ⎠

∑

∑

∑

WITS Lab, NSYSU.109

Derivation of Erlang CSubstituting for P0

[ ]1

0

1

0

1! 1 channels are busy

11 1 1! ! 1

1! 1!

C

r

k CC

k

C

C kC

k

CP C

Ck k

C

CC k

λµλµ λ λ

µ µ λµ

λµ

λ λ λµ µ µ

−

=

−

=

⎛ ⎞⎜ ⎟⎝ ⎠=

⎛ ⎞ ⎡ ⎤−⎜ ⎟ ⎢ ⎥⎛ ⎞ ⎛ ⎞⎝ ⎠ ⎢ ⎥+⎜ ⎟ ⎜ ⎟⎢ ⎥⎛ ⎞⎝ ⎠ ⎝ ⎠ −⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦

⎛ ⎞⎜ ⎟⎝ ⎠=

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

∑

∑

WITS Lab, NSYSU.110

Derivation of Erlang CWith A=λ/μ=λH, the Erlang C formula is given by:

[ ] 1

0

Pr channels are busy! 1

!

C

kCC

k

ACA AA CC k

−

=

=⎛ ⎞+ −⎜ ⎟⎝ ⎠

∑