1.1.1 The Service System

Consider that a single service‐class (e.g., telephone service) is accommodated to a loss system, say a transmission link, of capacity b.u. Each call arrives in the system according to a Poisson process with mean value and requires 1 b.u. to be serviced. If this bandwidth is available, then a call is accepted in the system and remains under service for an exponentially distributed service time, with mean value . Otherwise, when all b.u. are occupied, a call is blocked and lost without further affecting the system (e.g., a blocked call is not allowed to retry).

Now, let be the number of in‐service calls at the time instant . Since each call occupies 1 b.u. then also expresses the number of occupied b.u. at the time instant , i.e., . Assume that at time instant , (), the number of in‐service calls is , that is, ; in the teletraffic jargon, we say that the system is in state . Let denote the probability of being in state . Since calls arrive at random, based on (I.10) the system's state becomes at , if one of the following events takes place:

1. (a) and (no arrival or departure occurs in ).
2. (b) and (one new call arrives in ).
3. (c) and (one in‐service call departs from the system in ).

The probability of the first event equals . Note that the probability for a specific call to depart within is ; given that there are in‐service calls, one departure is possible due to the first or the second or the th call, and therefore the probability of one departure becomes the sum of probabilities: .

The probability of the second event equals , while the probability of the last event equals (because of the in‐service calls).

Therefore, because the above events are exclusive, we take:

(1.1)

or

(1.2)

In (1.2), when , the limit of the LHS defines the derivative of :

(1.3)

According to the notion of derivative, (1.3) shows the rate of changes of the instantaneous value of , which is of little interest. Instead, of great interest is to find the state probability in the steady state, that is, when the system operates normally for a long time period. Service systems normally have a steady state; this means that as . Then, the probability approaches a limiting value , which is constant over time and is determined through ( 1.3) when the LHS is zero:

(1.4)

where and .

Since (1.4) holds for all , we say that the system is in statistical equilibrium. To find the limiting probabilities , also called steady state probabilities, we solve ( 1.4) by applying the so‐called ladder method:

By adding these equations side by side, we obtain the following recurrent formula:

(1.5)

where is the offered traffic‐load in erl.

From (1.5), by successive substitutions, we relate to the probability that the system is empty, :

(1.6)

One more equation is needed between and to formulate a system of two equations with two unknowns. Since the system will always be in one of the states , we have:

(1.7)

that is,

(1.8)

By substituting (1.8) to (1.6), we determine :

(1.9)

which is the well‐known Erlang distribution.¹

1.1.2 Global and Local Balance

It is important at this point to interpret ( 1.4) and ( 1.5). Let us rewrite them as follows, while multiplying them by :

(1.10a)

(1.10b)

The LHS of (1.10a) shows the probability sum of two events while the system is in state : a call arrival () and a call departure (), which transfer the system out of state , obviously to the adjacent state or , respectively. Likewise, the RHS of (1.10a) shows the probability of moving to state from the adjacent states and , due to a call arrival and a call departure, respectively. Let us remove from this equation. Then, (1.10a) denotes the fact that rate‐out = rate‐in and is named the global balance (GB) equation of state .

Even more interesting is the interpretation of (1.10b), where the RHS shows the probability of moving up, from state to state , due to a call arrival, while the LHS shows the probability of moving down, from state to state , due to a call departure. When removing , this equation denotes the fact that rate‐down = rate‐up and is named the local balance (LB) equation between the adjacent states and .

Having removed from (1.10a), its graphical representation is shown in Figure 1.1 by the state transition diagram of the system. Two LB equations are presented (the first one between states and and the second one between states and ). Note at this point that the existence of GB between adjacent states does not guarantee the existence of LB. The opposite holds.

Equations (1.10) introduce the following methodology, called classical methodology, in analyzing a service system in equilibrium, i.e., in determining its steady state probabilities:

1. (i) Start with the state transition diagram of the system.
2. (ii) Write the GB equations.
3. (iii) Apply the ladder method to the GB equations in order to obtain simpler equations, usually the LB equations.
4. (iv) If steps (ii) and (iii) are too complicated, assume that LB equations hold and go on, but check for atopy.
5. (v) Consider the normalization condition.²
6. (vi) Solve the resultant linear system of equations of the steady state probabilities.

1.1.3 Call Blocking Probability

The most³ important question in a loss system is “What is the CBP?” or, equivalently, “What is the GoS?”. Call blocking occurs when the system is fully occupied, that is, CBP equals , the probability that the system is in state . From ( 1.9) we have

(1.22)

Equation (1.22) is the famous Erlang‐B formula also met in (I.5). As we discussed there (Example I.5), the closed form of the Erlang‐B formula is not appealing for large values of and , instead its recurrent form (I.8) is not only preferred but necessary.

It is worth mentioning at this point that, as has been investigated (e.g., [1–3]), ( 1.9) and consequently ( 1.22) are insensitive to the distribution of service time, and depend only on the mean holding time, , which is inherently included in the traffic‐load (). Besides, ( 1.22) refers to the proportion of time that all b.u. are occupied (i.e., the system is congested). This probability is named time congestion (TC) probability and can be measured by an outside observer. Due to the PASTA property, an inside observer sees the same probability, therefore ( 1.22) is also called call congestion (CC) probability or CBP.

Having found a relationship between , and GoS, i.e., , let us now provide graphs of them (Figure 1.3) in order to compare them with the qualitative graphs of Figure I.2. The first graph of Figure 1.3 presents the required system capacity versus the offered traffic‐load (for two certain values of GoS: 1% and 3%) and corresponds to the LHS graph of Figure I.2. However, the anticipated curve of Figure I.2 is hardly followed in Figure 1.3; the function is rather a straight line in a wide range of the presented values of traffic‐load. The second graph of Figure 1.3 presents CBP versus , for erl and erl, and corresponds to the middle graph of Figure I.2. Due to the reverse meaning of CBP,⁴ the convex curvature of the middle graph in Figure I.2 appears in the middle graph of Figure 1.3, as a concave (reverse) curvature. The third graph of Figure 1.3 presents CBP versus traffic‐load for two values of : 5 and 6 b.u. Herein, the corresponding qualitative graph of Figure I.2 (RHS) is followed pretty well.

1.1.4 Other Performance Metrics

• Utilization: The utilization, , is expressed by the average number of occupied b.u.:
  (1.24)

  Equation (1.24) verifies (I.34). Because of property (4) of traffic‐load, .

• Trunk efficiency: According to (I.35), the trunk efficiency is:
  (1.25)

  Since expresses traffic‐load per trunk, (property (3) of traffic‐load). Thus, again, . As Figure 1.4 shows, the trunk efficiency increases as the system capacity increases, for a certain GoS. This is called the large‐scale effect and leads to the conclusion that loss systems must be designed with the greatest possible capacity in order for trunks to be used efficiently. The latter happens because the larger the system capacity, the greater the carried traffic conveyed under a certain GoS.

• A low bound of : Since always, we have:
  (1.26)

  Equation (1.26) can be used for a fast evaluation of CBP measurements or CBP calculations, given that the offered traffic‐load estimation is correct. For instance, according to ( 1.26), if b.u. and erl, the anticipated CBP % at least. Note that the actual CBP value is . The interested reader may resort to [4] and the references therein for an in‐depth analysis of the lower and upper bounds on the Erlang‐B and Erlang‐C formulas.

1.2.1 The Service System

Let us now consider the multi‐service system or, as we call it, the Erlang multirate loss model (EMLM). A single link of capacity b.u. accommodates calls of different service‐classes under the CS policy. Each call of service‐class arrives in the system following a Poisson process with mean rate and requires b.u. to be serviced. If the requested bandwidth is available, then a call is accepted in the system and remains under service for an exponentially distributed service time, with mean . Otherwise, the call is blocked and lost, without further affecting the system. After service completion, the b.u. are released and become available to new arriving calls.

Let denote the number of in‐service calls of service‐class in the steady state, ) the corresponding vector of all in‐service calls of all service‐classes, and ) the corresponding vector of the required bandwidth per call of all service‐classes in the system. Because of the CS policy, the set of the system (state space) is given by (I.36). The product expresses the occupied link bandwidth in system state and plays a decisive role in CAC:

(1.27)

In terms of , the CAC is expressed as follows. A new call of service‐class that finds the system in state is accepted in the system if , where .

In any case, the best way to prove that a system has a PFS,⁵ is to find it!

1.2.2 The Analytical Model

1.2.2.1 Steady State Probabilities

In analyzing a service system, the first target is to determine the steady state probability . For the EMLM, it can be determined, obviously, by extending the PFS (1.29) of Example 1.6 to service‐classes, as follows:

(1.30)

where the normalization constant, which is determined through :

(1.31)

Of course, the PFS (1.30) must satisfy the set of GB and LB equations of the system. To verify it, follow the aforementioned classical methodology:

1. (i) Draw the state transition diagram and write the GB equations. Figure 1.7 shows the one‐dimensional state transition diagram of the EMLM when a general state is considered. Normally, the state transition diagram has as many dimensions (axes) as the number of service‐classes ; see the LHS of Figure 1.8. To be converted to a one‐dimensional diagram, we define the equivalent diagram at the RHS of Figure 1.8. This consideration is justified because both diagrams lead to the same GB equation (see 1.32). Thus, the GB equation (rate in = rate out) for state is given by:
   (1.32)
   where are the probability distributions of the corresponding states ; parameters validate a state transition through the following expressions:
   (1.33)

1. (ii) Assume that LB exists and check for atopy.

   Assuming the existence of LB between any adjacent states ( and , or and , for ), then the following LB equations (rate up = rate down) are extracted from the state transition diagram (Figure 1.8, LHS) (correspondingly):

   (1.34a)
   (1.34b)
2. (iii) Solve the resultant linear system of equations of the equilibrium probabilities, while considering the normalization condition. Check for atopy, again.

   It can be verified that the probability distribution has a PFS by substituting ( 1.30) into (1.34a) or (1.34b).

Having calculated the steady state probability , we proceed to determine the CBP. To this end, we denote by the admissible state space of service‐class : . A new service‐class call is accepted in the system, if, at the time point of its arrival, the system is in a state . Hence, the CBP of service‐class is determined by the state space , as follows:

(1.36)

Equation ( 1.35) provides the CBP of the EMLM based on a PFS. A recurrent form for the CBP calculation can be obtained by expressing ( 1.36), as follows:

(1.37)

where ; the values of can be determined recursively via:

(1.38)

For large values of the computational complexity of (1.38) is ⁶. A simpler formula follows with a computational complexity ,⁷ for the determination of the state probabilities of the system, when the system state is represented not by the number of in‐service calls of each service‐class, but by the total occupied b.u. in the link, where . This state representation is more effective when aimed at calculating the key performance metrics of the system, like CBP and link utilization. The unnormalized values of the link occupancy distribution, i.e., the probability that out of b.u. are occupied, are given by:

(1.39)

Equation (1.39), known in the literature as the Kaufman–Roberts⁸ recursion, is accurate and computationally efficient with an easy computer implementation. Because of this, it is the springboard to derive other more complex but efficient teletraffic models. In order for the values of ( 1.39) to become probabilities, they must be normalized through division by the sum of them, :

(1.40)

Note that denotes either a normalized or unnormalized value of the link occupancy distribution, but, in any case, it will be explicitly mentioned (unless it is clear), while denotes a normalized value of the link occupancy distribution.

Proof of ( 1.39): According to [5], let us consider the occupied link bandwidth , as a system state. Thus, the system can be seen either via state (multi‐dimensional system) or via the new (aggregate) state (one‐dimensional system). The link occupancy distribution is defined as:

(1.41)

where is the set of states in which exactly b.u. are occupied by all in‐service calls: (Figure 1.9).

The key point for the recursive calculation of is to associate the values of values with the “previous” values of . In other words, we have to relate the values of with the (“previous”) values of , since in state there are in‐service calls of service‐class , while in state there are in‐service calls (assuming that ). This relation will be achieved by the use of the LB equation (1.34a), as we will shortly show.

Since , we can write (1.41) as follows:

(1.42)

From the LB equation (1.34a), the product is determined by:

(1.43)

where the parameter is replaced by (another binary parameter) to denote that:

(1.44)

We take sums of both sides of (1.43) over to have:

(1.45)

Equation ( 1.43) has no meaning when ; since the RHS of (1.45) refers to state , the previous state (in which the LHS of ( 1.45) refers to) belongs to given that . This is expressed through the parameters . More formally, when , the LHS of ( 1.45) is written as:

(1.46)

and the set defines the state space . To individuate the case where , we introduce the following variable:

(1.47)

By using (1.47), and because of the definition ( 1.41), we can write (1.46) as follows:

(1.48)

From ( 1.45), ( 1.46), and (1.48), we obtain:

(1.49)

Based on (1.49), (1.42) is written as:

(1.50)

which is the Kaufman–Roberts recursion ( 1.39). For an alternative proof see Example 1.12.

Q.E.D.

According to [ 5], ( 1.39) can be used for arbitrary distributed service times. An interesting interpretation of ( 1.39) is that it stands for an LB equation of a birth–death process, in which is the birth rate of service‐class calls, is the corresponding death rate, and is the mean number of service‐class calls in state (Figure 1.10):

(1.51)

Indeed, (1.51) is derived from ( 1.49) because the RHS of ( 1.49) is written as follows:

(1.52)

Note that ( 1.39) can be derived from ( 1.51) by multiplying both sides of ( 1.51) by and summing over .

1.2.2.2 CBP, Utilization, and Mean Number of In‐service Calls

The following performance measures are determined based on ( 1.39):

• CBP: The determination of CBP of service‐class , is given by:
  (1.53)

  where is given by (1.40).

  Figure 1.11 depicts a helpful visualization regarding (1.53).

  

  Note that ( 1.53) refers to the TC probabilities of service‐class calls. These probabilities coincide with the CBP due to the PASTA property.

  Needless to say that in the case of one service‐class in the system, the CBP obtained by ( 1.53) coincides with the results of the Erlang‐B formula; hence, we name this model the Erlang multirate loss model.

• Utilization: The link utilization, , is calculated by:
  (1.54)
• Mean number of in‐service calls in state : The mean number of in‐service calls of service‐class in state is given (because of ( 1.51)) by:
  (1.55)

  Note that , if .

• Mean number of in‐service calls in the system: The mean number of in‐service calls for service‐class in the system, , is given by:
  (1.56)

1.3.1 The Service System

We consider again the multi‐service system of the EMLM, but with the following CAC. A new service‐class call is accepted in the link if, after its acceptance, the link has at least b.u. available to serve calls of other service‐classes. This service system is called EMLM under the BR policy (EMLM/BR). By properly selecting the BR parameters , we can achieve CBP equalization among service‐classes; this is the main target of the BR policy. Assuming that , then for CBP equalization the parameters are chosen so that , that is, , since it is reasonable not to reserve bandwidth against the service‐class which requires the maximum bandwidth per call. Obviously, due to CBP equalization, we avoid the CBP oscillations observed in the EMLM under the CS policy.

Figure 1.13 illustrates the case of a single link with that accommodates calls of two service‐classes with and b.u. To achieve CBP equalization we reserve b.u. in favor of calls of the second service‐class.

1.3.2 The Analytical Model

The basic characteristic of the BR policy is that the steady state probabilities cannot be calculated via a PFS. This is because LB between some adjacent states is destroyed (see the following example).

1.3.2.2 Approximate CBP Calculation based on the Roberts Method

In the EMLM/BR, the link occupancy distribution, q(j), is given in an approximate way by the following recursive formula [11]:

(1.64)

where:

(1.65)

This formula has a form similar to the Kaufman–Roberts recursion ( 1.39), and its existence is based on the assumption that, for a service‐class with , the mean number of service‐class calls in state , , is zero in all states which belong to the prohibitive space of this service‐class: . Thanks to this assumption, which is reflected in the variable , the one‐dimensional Markov chain of the system is transformed to an approximate reversible Markov chain,⁹ which leads to the recurrent formula (1.64). Markov chain reversibility is a strong indication of the existence of a PFS.

The CBP of service‐class , , is given by:

(1.66)

where is the normalization constant, given by ( 1.40).

The link utilization, , is given by ( 1.54).

The mean number of service‐class calls, in state , is determined by:

(1.67)

The mean number of service‐class calls in the system, , is given by ( 1.56).

1.3.2.3 CBP Calculation Recursively based on the Stasiak–Glabowski Method

When aiming at QoS equalization among service‐classes, the recursive CBP calculation of the EMLM/BR according to the Roberts method can be improved by the following method proposed by Stasiak and Glabowski¹⁰ [12]. The average number of service‐class calls, , in state is not zero but positive, and can be determined approximately in a recurrent way by:

(1.68)

where for the calculation of the corresponding EMLM system under the CS policy is assumed (i.e., the Kaufman–Roberts recursion ( 1.39)), while is a weight given by:

(1.69)

The weight determines the proportion of that is transferred in state by a call of service‐class (other than ), assuming that . Although the system cannot be in state due to an arriving call of service‐class (because of the BR policy), the system can be in state due to arriving calls of other service‐classes (calls of other service‐classes may coexist in previous states together with service‐class calls). Thus, when the system is transferred to state by a service‐class call, this call also transfers to state the population of service‐class . Therefore, the assumption that the average number of calls is positive even in a prohibitive state of a service‐class is more realistic compared to the Roberts assumption. Figure 1.16 illustrates the fact that calls from different service‐classes may contribute in transferring the population of service‐class to a prohibitive state . Consequently, given that in state the population of service‐class does exist, a backward transition to state is true.

Having determined the average number of calls in each state via (1.68), then the Roberts method is followed by replacing in the RHS of in ( 1.64) by , in order for the average number of calls of each service‐class in state to be determined:

(1.70)

The philosophy behind this method is that the approximated reversible Markov chain of the Roberts method is kept, but each state of the prohibited state space is now substituted by state . The Stasiak–Glabowski method is summarized in the following procedure:

1. Step 1: Assuming that the system is under the CS policy (instead of the BR policy), calculate via ( 1.39).
2. Step 2: Calculate the average number of service‐class calls in state , , according to ( 1.68) and (1.69).
3. Step 3: Determine for the EMLM/BR, as follows:
   (1.71)

   where is given by (1.70) and is given by ( 1.65).

4. Step 4: Determine the CBP of each service‐class according to ( 1.66).

1.4.1 The Service System

We consider again the multi‐service system of the EMLM and adopt a TH‐type CAC, as follows: A new call of service‐class k is accepted in the system, of C b.u., if:

1. (i) its bandwidth requirement, b_k b.u., is less or equal to the available link bandwidth
2. (ii) the number n_k of in‐service calls of service‐class k does not exceed a predefined threshold parameter, after its acceptance. Otherwise, the call is blocked and lost without affecting the system.

By definition a policy is called a TH policy if there exists a set of positive integers such that a service‐class call is accepted in the system when in state , if and only if the new system state fulfils the relations and [13].

1.4.2 The Analytical Model

1.4.2.1 Steady State Probabilities

Due to the fact that the TH policy is a coordinate convex policy (as the CS policy is) the steady state probabilities in the EMLM/TH have a PFS whose form is the same as that of the EMLM (only the definition of differs):

(1.72)

where and

For the determination of the unnormalized values of q(j), the following accurate and recursive formula can be used [14]:

(1.73)

where is the probability that b.u. are occupied, while the number of service‐class calls is or:

(1.74)

In (1.73) the fact that implies that:

1. (i) and therefore for and
2. (ii) is a blocking probability factor for service‐class calls.

The proof of ( 1.73) is similar to the proof of the Kaufman–Roberts formula (see e.g., [ 14]). The only difference is that ( 1.48) now takes the form:

(1.75)

due to the existence of the TH policy.

Intuitively, the form of (1.75) is expected, since the term is a factor that blocks a new service‐class call from being accepted in the system and therefore blocks the transition from state to state .

1.4.2.2 CBP, Utilization and Mean Number of In‐service Calls

The following performance measures can be determined:

• To determine the CBP of service‐class , , we consider two groups of macro‐states:
  1. (i) those where there is no available bandwidth to accept a new service‐class call; this happens when
  2. (ii) those where available bandwidth exists, i.e., but ; the latter implies that .

  The values of are given by:

  (1.76)

  where is the normalization constant.

• The link utilization is given by ( 1.54).
• The mean number of service‐class calls, in state , is given by:
  (1.77)
• The mean number of service‐class calls in the system is given by ( 1.56).

Equations (1.76) and (1.77) require knowledge of . The latter takes positive values when . Thus, we consider a subsystem of capacity that accommodates all service‐classes but service‐class . For this subsystem, we define , which is analogous to of ( 1.73):

(1.78)

We can now compute for , as follows:

(1.79)

In (1.79), the term is expected, since for states , the number of in‐service calls of service‐class is always .

The computational complexity of the EMLM/TH is in the order of , where ; for more details, see [ 14].

Note: In (1.78), each time that the calculation of requires knowledge of (this happens when , for , an extra subsystem is needed for the calculation of and via ( 1.78) and ( 1.79) (see also Section IV, pp. 12–69 in [ 14]). Instead of using subsystems, we may calculate for those values of that result in (in ( 1.78)), while for the rest values of we can use the following formula (which is based on the fact that the EMLM/TH has a PFS):

(1.80)

for and .

This method can be useful especially for systems with small to moderate state spaces (see Example 3.10). This topic remains open to investigation.

1.5.1 The Service System

According to ITU‐T, a fixed routing network is a network in which a route¹¹ providing a connection between an originating node and a destination node is fixed for every service‐class (or for every traffic flow of the same service‐class). Let us consider that a fixed routing network consists of links. Each link has a fixed capacity of b.u. The network accommodates calls of service‐classes under the CS policy. Calls of service‐class follow a Poisson process with rate , require b.u. and have a generally distributed service time with mean . Let be the fixed route of service‐class calls in the network, where . A call of service‐class is accepted in the network if its b.u. are available in every link , otherwise the call is blocked and lost.

1.5.2 The Analytical Model

Let be the number of in‐service calls of service‐class in the steady state of the system and be the corresponding steady state vector of all service‐classes in the fixed routing network. If is the set of service‐classes whose calls are accommodated in link , i.e., , then the state space of the system is given by:

(1.81)

1.5.2.1 Steady State Probabilities

The steady state probabilities have a PFS whose form is the following [ 13,15]:

(1.82)

where is the normalization constant, and is the offered traffic‐load of service‐class calls.

If we denote by the occupied b.u. of link , where , and is the corresponding vector of the entire fixed routing network, then the unnormalized values of the occupancy distribution in the fixed routing network are given by the following L‐dimensional accurate recursive formula [ 15]:

(1.83)

where is the th row of a () matrix (routing table), and shows the route (sequence of links) for the th service‐ class.

Note: In the case of a single link, (1.83) becomes the Kaufman–Roberts recursion ( 1.39).

1.5.3 CBP Calculation by the RLA Method

1.5.3.1 A Fixed Routing Network Supporting a Single Service‐class

We assume that a fixed routing network supports K different traffic flows of a single service‐class requiring 1 b.u. per traffic flow, that is, . Traffic flows are distinguished by the different sequence of links traversing the various end‐to‐end connections in a fixed routing network. Let be the total offered traffic‐load to a link , where :

(1.88)

where is the set of traffic flows utilizing link .

The CBP of traffic‐flow , can be upper bounded by the following product (based on the Erlang‐B formula) [19]:

(1.89)

This product‐bound provides a good CBP approximation only if the number of links used by traffic flows in the network is small. If not, this bound is unreliable, for example consider only one traffic flow traversing links of the same capacity . Then, from (1.89), we have [ 13]:

(1.90)

where for every link .

The bound of (1.90) approaches unity when increases, although for every link and, consequently, for the entire fixed routing network.

A better CBP approximation is achieved by reducing the offered traffic‐load so that blocking in the other links (excluding ) is taken into account. Specifically, this is done by substituting in (1.88) the term with , where the reduced factor is the probability that there is at least 1 b.u. available in every link of the route . Denoting the approximate CBP in link by , we obtain:

(1.91)

Assuming that blocking is independent from link to link (an assumption that is incorrect), we have [ 13]:

(1.92)

The combination of (1.91) and (1.92) gives the following fixed‐point equation¹² for the approximate CBP determination in link l:

(1.93)

Assuming again that blocking is independent from link to link, we approximate the CBP of traffic flow k, , as follows [ 13]:

(1.94)

The combination of (1.93) and (1.94) constitutes the RLA method (or the so‐called Erlang fixed point equation, see e.g., [20]) for fixed routing networks that support traffic flows of a single service‐class.

Note that the fixed point equation has a unique solution (see e.g., Theorem 5.9 in [ 13]) which satisfies the bound:

(1.95)

This solution can be obtained via a simple method that relies on repeated substitutions, as the following example shows.

1.5.3.2 A Fixed Routing Network Supporting Multiple Service‐classes

Consider now that service‐classes are accommodated in a fixed routing network of links, according to the service system described in Section 1.5.1. Suppose that a service‐class traverses a link of capacity b.u. and experiences there CBP, . can be determined approximately by:

(1.96)

where it is assumed that the offered traffic‐load of service‐class to the whole fixed routing network is the same as the offered traffic‐load of service‐class in link ; is the unnormalized probability of having occupied b.u. in this link (calculated by the Kaufman–Roberts recursion, ( 1.39), over all ) and is the corresponding normalization constant.

The CBP expression (1.96) can be improved by considering that the offered traffic‐load of a service‐class to a link is actually reduced when traversing through a sequence of links. Let us denote by the improved CBP of service‐class in link . Based on ( 1.92), the offered traffic‐load is reduced to . Then, is given by [ 13]:

(1.97)

Based on (1.97), the approximate CBP calculation of service‐ class , in the entire route , is given by:

(1.98)

The combination of ( 1.97) and (1.98) constitutes the RLA method for fixed routing networks that support calls of different service‐class. The values of can be obtained via repeated substitutions as Example 1.27 shows.

The extension of the RLA method to include different service‐ classes appears in the literature as the knapsack approximation [21]. This term is justified by the fact that the EMLM resembles the stochastic knapsack/problem in combinatorial optimization [22]. The knapsack approximation does not always lead to a unique solution (for an analytical example see [ 13]), however in most cases it approximates CBP quite satisfactorily.

1.6.1 The Erlang‐B Formula

Although the Erlang loss model is now obsolete, since it is applicable to single service‐class systems only, it is still useful when a network operator wants to guarantee a specific QoS for a service (of streaming traffic) and, to achieve it, reserves a certain amount of bandwidth, b.u. for this service (not recommended, because of no statistical multiplexing gain [24]). For the applicability of the Erlang‐B formula ( 1.22) when a call requires a service rate b.u., we have to consider a system capacity of and traffic‐load : . Thus, we transform the system as if it were . When the system capacity is a real number, is determined through the gamma function [25]:

(1.103)

1.6.2 The Erlang‐C Formula

For the Internet, more interesting is the case of the Erlang‐C formula (1.18). As has been investigated (e.g., [26], [27]), the Erlang‐C formula can provide an upper bound of the congestion probability in a lightly loaded link of the Internet with b.u., when a max–min fairness policy is applied among traffic flows sharing the link (see [28] and [29]). Assume that we can distinguish different peak rates (in increasing order) among the flows traversing the link. Let be the offered traffic‐load of flows which corresponds to peak rate ; then, the overall traffic‐load is (for a steady state).¹³ If is the number of active flows (calls) with peak rate , then congestion occurs when . In this case, when a fair rate is determined based on the max–min fairness policy, flows of higher rates (say, all flows with peak rates from to , ) must reduce their bandwidth to the fair rate (so that ) in order for the total rate to satisfy the link bandwidth capacity . The proportion of time where any active flow of rate would suffer loss or has to reduce its bandwidth should be less than a targeted value (i.e., the GoS); this proportion of time is defined as the rate‐ congestion probability¹⁴ [ 26]:

(1.104)

Equivalently, the rate‐ congestion probability is the probability that the max–min fair rate is less than (because is determined so that the system capacity is not violated). This probability increases when all higher peak rates are reduced to (or all lower peak rates increase to ), while keeping constant the overall traffic‐load . When the peak rates of higher peak rates are reduced to , the number of flows increases to (suppose for all ), so that the overall traffic‐load remains constant. Thus, the corresponding probability of rate‐ congestion also increases:

(1.105)

Intuitively, the max–min fair rate of the original system cannot be less than that of the new system (since and higher peak rates have been reduced in the new system).

The worst case of the distribution of the peak rates in the link for the rate‐ congestion probability under the max–min fair policy is the case where all flows have an equal peak rate . In that case, the rate‐ congestion probability cannot be increased since the max‐min fair rate cannot be reduced, and therefore it becomes an upper bound of the congestion probability. The total number of flows (each flow with the same peak rate bps) offering a total traffic‐load in the link of capacity (bps) resembles a FIFO queuing system with no loss (if it is lightly loaded), and thus can be modeled as . Specifically, if is an integer and , then the probability of congestion (i.e., the proportion of time where ) is given by the Erlang‐C formula ( 1.18) when the system capacity is and the offered traffic‐load is . Thus, we have:

(1.106)

The applicability of the Erlang‐C formula in the Internet fits well to the IntServ resource (bandwidth) allocation strategy, but not to the DiffServ strategy (see Section I.14). Max–min fairness can be implemented through TCP congestion control.

1.6.3 The Kaufman–Roberts Recursion

Applications of the EMLM (Kaufman–Roberts recursive formula ( 1.39)) in contemporary communications networks are numerous. Its applicability to integrated services digital networks (ISDN) or global system for mobile (GSM) communications networks is straightforward and there is no need for additional explanation.

In optical networks of wavelength division multiplexing (WDM) technology, each fiber supports multiple communication channels, each one operating at a different wavelength. In view of the fact that only a small number of network users necessitate the entire bandwidth of the channel (e.g., up to bps data rate), the network mostly supports traffic demands with data rates significantly lower than the full wavelength capacity. Therefore, the channel bandwidth is divided into lower sub‐rate units (called traffic grooming), while traffic streams use one or multiple of these units, i.e., b.u. Considering a single link and a certain capacity (in b.u.) in each wavelength of the link supporting several service‐classes, the application of ( 1.39) for the calculation of the occupancy distribution of each separate wavelength is also straightforward under the Poisson traffic assumption [30].

Let us concentrate now on wideband code division multiple access (WCDMA) wireless networks like the Universal Mobile Telecommunication System (UMTS) in Europe [31]. In a WCDMA cell, all users transmit in the same frequency band by using (pseudo) orthogonal codes to have their signals separated. The basic idea is that a user considers interference (noise) all other signals of all other users. The interference increases as the number of users increases, whereas the cell capacity of the uplink decreases; thus, the applicability of the EMLM is not straightforward but possible. We need to interpret the several sources of noise in terms of the EMLM (see also Section 2.11). The maximum cell load () can be seen as the system capacity , while the required bandwidth per call corresponds to the occupied cell's resources per call/user (load factor, ). The latter is determined by the bit‐error‐rate (BER) parameter, ¹⁵ and the transmission bit rate of the corresponding mobile station [32], [33]:

(1.107)

where is the chip rate of the WCDMA carrier.¹⁶

Since discretization is necessary to apply ( 1.39), this is achieved through the introduction of a cell load unit : , . The controls the granularity of the state space of the system, which increases as decreases. On the other hand, the smaller the , the better the analytical results (CBP on the uplink) we obtain. For a thorough study, several other parameters (such as local blocking (expressing the fact that a call may be blocked due to noise even if the cell accommodates a very few number of users), user activity, interference from other cells) must be incorporated in the EMLM for WCDMA networks [ 32]. Having understood the applicability of the EMLM to WCDMA systems, then it can readily be applied to more complex systems like the one presented in [34].

We concentrate now on the downlink of an orthogonal frequency division multiplexing (OFDM)‐based cell that services calls from different service‐classes with different traffic description parameters and consequently different QoS requirements. The cell has subcarriers and let and be the average data rate per subcarrier, the available power in the cell and the system's bandwidth, respectively. Let the entire range of channel gains or signal‐to‐noise ratios per unit power be partitioned into consecutive (but non‐overlapping) intervals and denoted as the average channel gain of the interval. Considering subcarrier requirements and average channel gains, there are service‐classes. A newly arriving service‐class call requires subcarriers in order to be accepted in the cell (i.e., the call has a data rate requirement ) and has an average channel gain . If these subcarriers are not available, the call is blocked and lost. Otherwise, the call remains in the cell for a generally distributed service time with mean . To calculate the power required to achieve the data rate of a subcarrier assigned to a call whose average channel gain is , we use the Hartley–Shannon theorem: . Assuming that calls follow a Poisson process with rate and that is the number of in‐service calls of service‐class , then the system can be described as a multirate loss model with a PFS for the steady state probabilities [35]:

(1.108)

where , is the state space of the system, is the normalization constant, and is the offered traffic‐load (in erl) of service‐class calls.

Note that the derivation of (1.108) requires that and are integers (which is generally not true). In order to have an equivalent representation of the constraint , we multiply both sides of this expression by a constant to get , where and are integers [ 35]. Thus, without loss of generality, we assume that and are integers. Now, let (i.e., ) be the occupied subcarriers and (i.e., ) the occupied power in the cell. Then, it is proved in [36] that there exists a recursive formula, which resembles the Kaufman–Roberts recursion, for the determination of and consequently all performance measures.

Finally, it is worth mentioning a remarkable application of the EMLM on smart grid, which is an example showing the wide applicability range of the model. To control energy consumption, all appliances are connected to a central controller. Each appliance has a power demand (in power units), while the appliances are distinguished into different types according to the demand. For a specific type of appliance, the power demand arrival processes (to the controller) can be assumed Poisson. The controller activates each power demand request upon arrival, if it is possible, given that the total amount of power (capacity in total power units) that can be distributed to the appliances is limited. Each appliance operates for an operating time (depending on its type), which is generally distributed. Assuming that the amounts of power can be discretized, the analogy between a communication link and the smart grid case becomes obvious: the total amount of power corresponds to the link bandwidth capacity, the various types of appliances to service‐ classes, the power demand of a specific appliance to the required bandwidth per service‐class call, the arrival process for power request to the call arrival process, and the operating time of appliances to the service time of calls. In this way, the probability distribution of the total power units devoted to the appliances can be determined through the EMLM. Alternatively, the EMLM can be used as a tool to calculate the total amount of power needed for guaranteeing power demands under a specific GoS per type of appliances [37].