These notes relate to the traffic functions available in the Crome Formula Editor.[4]

• What is Erlang?
  
• Grade of Service (GOS)
  
• Exponential Holding Time
  
• Number of Sources
  
• Disposition of Blocked Calls
  
• CROME Basic Erlang B Formulas
  
• CROME Extended Erlang B Formulas (with Retry)
  
• CROME Erlang C Formulas
  

Erlang is a dimensionless unit of a traffic load, where 1 Erlang equals 1 call-hour per hour, or 3600 call-seconds in one hour.  The traffic theory based on Erlang B and C is developed by a Danish scientist, A. K. Erlang.  The number of Erlangs per busy hour may be calculated as  Erlangs = (Calls/Busy Hour)(Mean Calls-Holding Time).

Grade of Service is a measure of the probability that a percentage of the offered traffic will be blocked or delayed.  Grade of Service is commonly expressed as the fraction of calls or demands that fail to receive immediate service (blocked calls), or the fraction of calls that are forced to wait longer than a given time for service (delayed calls).

Call holding time is the length of time during which a traffic source engages a traffic path or channel.  Frequency distribution of typical voice call holding times shows negative exponential curves.  This is true from both theoretical and statistical observation stand point of view.  In other words, shorter call holding times (e.g. less than three minute call) are much more frequent than longer call holding times (e.g. 10 minute calls).

Number of sources that can make a demand for service can have bearing on the service that these sources can obtain.  If there is only one channel and one source, it is obvious that the probability of blocking or delay is zero.  However, when the number of sources start to grow the probability of congestion is no longer zero.  While keeping the same total traffic load, the effect of adding sources diminish rapidly at a point where there is a negligible difference in the probability of congestion.  In most of the applications in telephone systems the number of sources compared to the servers is large.  In abc of the Telephone, the author assumes "infinite source" for all applications except for cases where the number of sources is only 10 times the number of the servers.  In these cases, modeling based on the "finite sources" is used such as Engset or Binomial, or Delay models.

There are three common ways in which a system disposes a blocked call.  The system can dispose a call in one of the following three ways:

1.        Calls Cleared - If a server is not found, blocked calls are cleared or lost.

2.        Calls Held - If a server is not found, the calls waits for an interval of time duration equaling to its holding time and gets cleared if a server is not found.

3.        Calls Delayed - If a server is not found, the call waits in a queue until a server is available to serve the call.

Traffic Formula Summary

When to use the Erlang B and Erlang C formula:

When calls are blocked, blocked calls can either be delayed (or queued), or simply lost or cleared out of the system.  If the traffic theory is applied to systems such as Trunked Radio System, the Erlang C formula is used because blocked calls are queued by the trunking controller.  If the traffic theory is applied to systems such as paging, telephone or cellular telephone systems, the Erlang B formula is used because the blocked calls are cleared.

Erlang B

The Erlang B formula is based on the assumptions that there are infinite sources, blocked calls are cleared, and constant or exponentially distributed holding times. 

where

A=Total traffic offered in Erlang

N=Number of servers in a full availability group

P=Probability of loss

Erlang C

The Erlang C model is based on following assumptions: infinite sources, blocked calls are queued, and holding times are exponentially distributed.  Another assumption is that waiting calls are served in the order in which they arrive in the waiting line (first-in-first-out).  Exponentially distributed means that the number of calls having increasingly longer duration is proportionately decreasing.  This means that the number of calls making longer calls (e.g., 30 minutes) is much smaller than the number of calls making shorter calls (e.g., 5 minutes).

In queuing systems, several terms are used to designate  the grade of service.

Average delay of all calls -- Indicates the average time that offered calls must wait for a connection to be established.  This includes those calls that have zero delays.

Average delay of delayed calls -- Indicates the average time that delayed calls (delay greater than zero) must wait for a connection to be established.

Probability of delay -- Indicates the probability that an offered call is not handled immediately.

Probability of delay being exceeded -- Indicates the probability that an offered call must wait longer than a specified time for establishment of a concern.

where

A=Total traffic offered in erlangs

N=Number of servers in a full availability group

P(>0)=Probability of delay greater than 0

The probability of delay greater than t is:

The average delay, D1, on all calls is:

The average delay, D2, on calls delayed is

The Erlang B traffic model is used by telephone system designers to estimate the number of lines required for PSTN connections (CO trunks) or private wire connections.  CROME has three basic functions that will calculate any one of the three primary variables (from the other two) used in this model:

The functions gos (erlangs,Np), nJustified(gos,erlangs), and erlangsGosN(gos,Np) are the three (3) basic functions in CROME that implement Basic Erlang B formulas.  There are three posisble unknown values to calculate (and thus three CROME equations).

1.        Busy Hour Traffic (erlangs), calculated via
erlangsGosN(gos,Np)

2.        Grade of Service (gos) calculated via
gos(erlangs,Np)

3.        Number of Justifiable Servers (Np) that can be justified is (i.e. required channels) calculated via
nJustified (gos,erlangs)

Busy Hour Traffic (or erlangs) is the number of hours of call traffic there are during the busiest hour of operation of a telephone system. The erlangs argument is typically carried traffic (Ac) during a busy hour.  carried traffic is the traffic measured on the Network.  Offered traffic (Ao) is equal to carried traffic (Ac), if 100% of the calls are serviced

Grade of Service (or gos) is the blocking or failure of calls due to an insufficient number of servers or lines being available.  E.g. 0.03 mean 3 calls blocked per 100 calls attempted or a 3% rate.

The Number of Servers (or Np) is the number of lines, trunks, channels or servers.  For an MSC this will typically be the number of lines (i.e. DS0s) in a trunk group.

Examples

Given various erlangs from 13.75 down to 10 (i.e. Ac or carried traffic), a desired grade of service of 20% or gos=0.2, CROME will calculate the number justifiable servers or DS0s, based upon the carried traffic (Ao), note the probabilities of retry is always 100% (Refer to  the next section if you wish to lower the retry probability).

The same table with the target gos=0.1 (or 10%).

The same table with the target gos=0.01 (or 1%).

The Extended Erlang B traffic model (i.e. with Retry) is used by telephone system designers to estimate the number of lines required for PSTN connections (CO trunks) or private wire connections taking into account retries.  This traffic model takes into account the additional traffic loading caused by blocked callers trying to immediately call again if their calls are blocked.  This traffic model may be used where no overflow facilities are available from the trunk group being designed.

The functions Aoff (Ac,Np,W), and nJustifiedAc(gos,Ac,Np,W) are the two (2) basic functions in CROME that implement Erlang B formulas with respect to retry.  The retry probability in CROME is denoted by W where 0 <= W <= 1.  The value W=1 indicates 100% retry.   WARNING THE SENSE OF 'W" SEEMS WRONG - looking at the tables I imagine W=1 is a 0% retry, and W=0 is a 100% retry!

Offered Traffic (Ao), calculated via

Aoff (Ac,Np,W)

As noted before offered traffic (Ao) is equal to carried traffic (Ac), if 100% of the calls are serviced, thus it follows that that for a 100% retry or W=1:

Ac = Aoff (Ac,Np,1.0)   for all values of Np from 1…infinity

The formula for determining offered traffic from carried traffic is as follows:

Ao = Ac * (1 - (W * gos(Ac,Np)) ) / ( 1 - gos(Ac,Np) )

Number of Servers (Np) that can be justified is (i.e. required channels) calculated via:

nJustifiedAc(gos,Ac,Np,W)

Examples

Given erlangs=10 (i.e. Ac or carried traffic), Np=9 i.e. the number of working servers or DS0s, the offered traffic (Ao) at various probabilities of retry determined by CROME is as follows:

The number of justifiable servers (Np) via the carried traffic (Ac) taking into account the same probability of retries (W) would be as follows:

When comparing this example to that of the Basic Erlang B functions, it should be apparent that an equivalence between the functions Aoff, nJustifiedAc, and nJustified exists i.e.

nJustifiedAc(gos,erlangs,Np,W) = nJustified (gos, Aoff(erlangs,Np,W))

This section is in preliminary draft to be added at a later date