We will first start with the simple case of a single network link (ignoring routing and control), i.e., to understand the following functional relation:
Call blocking =F(traffic load, capacity). (11.2.1)
Voice telephone networks operate on the following basic principle: there is a finite amount of capacity (bandwidth) and each arriving call must be allocated dedicated bandwidth for the duration of the call; if bandwidth for this call is not available, the call must be blocked. Thus, the user is required to retry when blocking occurs. Such systems are also referred to as loss systems.
The relationship between call arrival and blocking and capacity is an important traffic engineering issue in the voice telephone network. A key result in this regard is attributed to A. K. Erlang for his seminal work on how to compute blocking, almost a century ago. We need to explain a few things before we are ready to present his results.
For the purpose of this discussion, we will consider a network link in which calls are arriving on either end of the link destined for the other end of the link; this also reflects the fact that call bandwidth in the voice telephone network is bidirectional.
Call arrivals in the telephone network are random. However, to make it simple, we will assume temporarily that calls arrive in a deterministic fashion and that we are considering only a single voice circuit. First, suppose that calls arrive in a deterministic fashion at the start of an hour and the user talks for exactly an hour. Thus, one user occupies the circuit for an hour and no one else can use it. Now suppose the user talks for only 10 min and then hangs up. The circuit is free for others to use for the rest of the hour. In fact, if another user arrives at that instant and occupies the circuit for, say an additional 10 min, then a third user can start using the circuits 20 min into the hour. Thus, if we slice the length of the calls to fixed 10-min windows, the system can accommodate six calls; that is, this looks like the system can handle six arrivals per hour (each of 10 minutes’ duration) as opposed to one arrival per user (using the entire hour for talking), while in either case just one circuit was considered! Simply put, this intuitively says that an increase in the call arrival rate does not necessarily mean that we need more circuits (or in general bandwidth) since the call duration is also a critical factor.
From the above illustration, it is clear that we need to consider call arrival as well as call duration to understand the notion of traffic. However, note that both call arrival as well as call duration are actually random, not deterministic as we have used in the simple illustra- tion above. In Table 11.1, we have listed the call arrival time, duration, and end time of seven randomly arriving calls; in Figure 11.1, we have plotted this information in terms of num- ber of busy circuits; note that the number of busy circuits also accordingly has a nonuniform behavior. In analysis, however, we use the average call arrival rate and the average duration.
However, to account for any random event, we need to see what type of statistical distri- bution is appropriate. It has been found that interarrival time between calls is exponentially distributed, which is equivalent to saying that the call arrival follows the Poisson process (see Appendix B.10); thus, we sometimes loosely refer to call traffic as Poisson traffic. Further- more, the call duration time is found to follow the exponential distribution. Given the av- erage call arrival rate and the average duration of a call, a good way to capture the traffic demand volume for telephone traffic is to consider the product of these two terms. Ifλis the
TA B L E 11.1 Call information.
Call Start Time Duration End Time
Number (in sec) (in sec) (in sec)
1 2.3 145.3 147.6
2 6.7 128.8 135.5
3 45.2 18.4 63.6
4 62.2 512.5 574.7
5 73.2 96.2 169.4
6 94.1 1045.7 1139.8
7 196.6 15.2 211.8
F I G U R E 11.1 Number of busy circuits (trunks) as calls arrive and leave over time (shown up to 200 sec) for calls listed in Table 11.1.
average call arrival rate, andτ is the average duration of a call, the traffic demand, referred to as offered load or offered traffic, is given by
a=λτ. (11.2.2)
This dimensionless quantity is given the unit name Erlang (or Erl), in honor of A. K. Erlang.
This quantity is hard to visualize since it is a product of two terms. However, there is a nice physical interpretation: this quantity refers to the average number of ongoing (busy) calls offered to a network link if we were to assume the link to have infinite capacity. In practice, we do not have infinite capacity; thus, there is always a chance that (some) calls will be blocked.
This also means that we need to consider another entity called carried load or carried traffic to refer to traffic that is carried (not blocked) due to finiteness of capacity. It is easy to see that in the case of infinite capacity, carried load is the same as the offered load.
A. K. Erlang’s work is profound in that he determined how to compute call blocking probability when average offered load and capacity are given. Ifcis the capacity of a link in terms of number of voice circuits, then call blocking for offered load a is given by the following Erlang-B loss formula:
F I G U R E 11.2 Call sequence and circuit occupancy for calls listed in Table 11.1.
B(a,c)= a
c
cc!
k=0ak k!
. (11.2.3)
This then is the relationship we noted in a generic way in Eq. (11.2.1). Several different aspects can be learned from the above formula. We present two illustrations.
Example 11.1 Erlang blocking illustration.
Consider a T1-link (1.54 Mbps) where calls are offered. Since each digitized voice circuit requires 64 Kilobits per second (Kbps) (=K), the bandwidth in terms of call units isc=C/K whereCis the raw data rate of the link; in the case of a T1-link, we havec=24(≈1.54/.064) voice circuits. Suppose that a load of 20 Erl is offered to this link. Then, the call blocking probability obtained using Eq. (11.2.3) is 6.6%.
We note the following important aspects: (1) it is possible to offer a load higher than 24 Erl to a link of capacity 24 since this is a loss system, and there is still a chance that some calls will go through; in fact, the call blocking probability for an offered load of 24 Erl to a capacity of 24 units, is 14.6%; and (2) blocking also benefits from scaling due to nonlinearity of the blocking formula; for example, if we increase the capacity 10 times to 240 circuits and the load 10 times to 240 Erl, the blocking drops to 4.9% (from 14.6%).
It is also important to understand what happens to load and blocking when the average call duration is changed. The following example illustrates that.
Example 11.2 Change in average holding time and impact on blocking.
Recall that offered load is a combination of two parameters: average call arrival rate and average call duration. Suppose that the offered load is 15 Erl; this load offered to a link with 24 units of bandwidth results in 0.8% call blocking. If the average call duration is 3 min, the average call arrival rate is 5/min. Suppose in a network the average call arrival rate stays the same (at 5/min), and the average call duration increases from 3 to 6 min; we can then see that the newly determined offered load is 30 Erl (=5×6); then, call blocking for 30 Erl offered to a link of 24 voice circuits is 27.1%! That is, without any increase in the call arrival rate, the offered load increases if the average call duration increases, thus impacting the call blocking
rate.
So far, we have talked about traffic, but we have not talked about directional traffic while telephone trunks are most often bidirectional. In the following, we illustrate what it means to be directional traffic in terms of blocking.
Example 11.3 Bidirectional trunks, directional traffic, and blocking.
Consider the simple case of a network link where traffic arrives on either end destined for the other end; thus, whenever a circuit is found free, either side can grab it. The question then is: what is the blocking observed by each side?
Let us consider this problem with some numbers. Let the number of circuits on a link be 100 between two nodes 1 and 2 connected directly by a bidirectional trunk group. Let the offered load from node 1 to node 2 be 85 Erl and let the offered load from node 2 to node 1 be 5 Erl. You might be misled to think that node 1 to node 2 should face more blocking since it has more traffic than the other direction (in fact, 17 times more). Actually, both sides will face the same blocking. Due to Poisson arrival of traffic and since either side can grab a circuit as long as there is one available without any preferential treatment, the sum traffic remains Poisson (see Appendix B.10) and we can sum up the loads. Thus, this situation is equivalent to the case of a total of 90 Erl being offered to a link with 100 circuits; the call blocking from Erlang-B loss formula is then found to be 2.69%.
In fact, if 45 Erl is offered from one side and another 45 Erl from the other side, the block- ing value will remain the same since the total offered load is still 90 Erl.
Remark 11.1. Blocking with multiple traffic classes.
The scenario for two traffic classes follows the same principle described in the above example. That is, if there are two different traffic classes using the same link, both classes will observe the same blocking value regardless of each one’s specific load as long as the per-call bandwidth for both classes is the same. If the per-call bandwidth is different for different traffic classes, the situation changes; in fact, the Erlang-B formula is not applicable any more.
This situation will be discussed later in Section 17.6.
In the examples discussed so far, we have computed results using the Erlang-B loss for- mula (11.2.3); a closer look reveals that it includes both factorial (c!) and exponential (ac) terms, which can cause numerical difficulty for large numbers. Below, we describe a simple way to compute this formula.
11.2.1 Computing Erlang-B Loss Formula
It may be noted that the Erlang-B loss formula given in Eq. (11.2.3) can also be expressed through the following recurrence relation:
B(a,c)= aB(a,c−1)
c+aB(a,c−1) (11.2.4)
with the initial conditionB(a,0)=1.
If we writeB(a,c)=1/d(a,c), then we can rewrite the above recurrence relation as fol- lows:
d(a,c)=c d(a,c−1)/a+1.
Using this result, we can develop an iterative algorithm, as given in Algorithm 11.1, for computing call blocking when the offered load and the number of circuits are given.
While this is the basic idea of the algorithm, in an actual implementation, some nu- merical round-off issues should also be addressed for extreme cases of load and capac- ity.
A L G O R I T H M 11.1 Computing blocking using Erlang-B loss formula procedure erlangb (a,c)
if (a≤0orc<0) return ’input not valid’
d=1
fori=1, . . . ,cdo d=i∗d/a+1 endfor
b=1/d return(b) end procedure