4
Multirate Elastic Adaptive Retry Loss Models
In this chapter we consider multirate loss models of random arriving calls not only with elastic bandwidth requirements upon arrival but also elastic bandwidth allocation during service. Calls may retry several times upon arrival, requiring less bandwidth each time, in order to be accepted for service. If call admission is not possible with the last (least) bandwidth requirement then bandwidth compression is attempted.
4.1 The Elastic Single‐Retry Model
4.1.1 The Service System
In the elastic single‐retry model (E‐SRM), we consider a link of capacity 
b.u. that accommodates elastic calls of 
service‐classes. Calls of service‐class 
follow a Poisson process with mean arrival rate 
and have a peak‐bandwidth requirement of 
b.u. and an exponentially distributed service time with mean 
. To introduce bandwidth compression, we permit the occupied link bandwidth 
to virtually exceed 
up to a limit of 
b.u. Let 
be the occupied link bandwidth, 
, when a new service‐class 
call arrives in the link. Then, for call admission we consider the following cases:
- (a) If
, no bandwidth compression takes place and the call is accepted in the link with 
b.u. - (b)
If
, then the call is blocked with 
and retries immediately to be connected in the link with 
. Now if:
- b1)
, no bandwidth compression occurs and the retry call is accepted in the system with 
and 
, so that 
, - b2)
, the retry call is blocked and lost, and - b3)
, the retry call is accepted in the system by compressing its bandwidth requirement 
together with the bandwidth of all in‐service calls of all service‐classes. In that case, the compressed bandwidth of the retry call becomes 
, where 
is the compression factor, common to all service‐classes. Similarly, all in‐service calls, which have been accepted in the link with 
(or 
), compress their bandwidth to 
(or 
) for 
. After the compression of all calls, the link state is 
. The minimum value of the compression factor is 
.
- b1)
Similar to the E‐EMLM, when a service‐class 
call, with bandwidth 
(or 
), departs from the system, the remaining in‐service calls of each service‐class 
expand their bandwidth in proportion to their initially assigned bandwidth 
(or 
). After bandwidth compression/expansion, elastic service‐class calls increase/decrease their service time so that the product service time by bandwidth remains constant.
Similar to the SRM, the steady state probabilities in the E‐SRM do not have a PFS, since LB is destroyed between adjacent states (see Figure 4.1). Thus, the unnormalized values of 
can be determined by an approximate but recursive formula, as presented in Section 4.1.2.
Figure 4.1 The state space 
and the state transition diagram (Example 4.1).
Table 4.1 The state space 
and the occupied link bandwidth (Example 4.1).
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
| (before compr.) | (after compr.) | (before compr.) | (after compr.) | ||||||||
| 0 | 0 | 0 | 1.00 | 0 | 0 | 2 | 0 | 0 | 1.00 | 2 | 2 |
| 0 | 0 | 1 | 1.00 | 2 | 2 | 2 | 0 | 1 | 1.00 | 4 | 4 |
| 0 | 0 | 2 | 1.00 | 4 | 4 | 2 | 0 | 2 | 0.67 | 6 | 4 |
| 0 | 0 | 3 | 0.67 | 6 | 4 | 2 | 1 | 0 | 0.80 | 5 | 4 |
| 0 | 1 | 0 | 1.00 | 3 | 3 | 3 | 0 | 0 | 1.00 | 3 | 3 |
| 0 | 1 | 1 | 0.80 | 5 | 4 | 3 | 0 | 1 | 0.80 | 5 | 4 |
| 1 | 0 | 0 | 1.00 | 1 | 1 | 3 | 1 | 0 | 0.67 | 6 | 4 |
| 1 | 0 | 1 | 1.00 | 3 | 3 | 4 | 0 | 0 | 1.00 | 4 | 4 |
| 1 | 0 | 2 | 0.80 | 5 | 4 | 4 | 0 | 1 | 0.67 | 6 | 4 |
| 1 | 1 | 0 | 1.00 | 4 | 4 | 5 | 0 | 0 | 0.80 | 5 | 4 |
| 1 | 1 | 1 | 0.67 | 6 | 4 | 6 | 0 | 0 | 0.67 | 6 | 4 |
To facilitate the recursive calculation of 
, we replace 
by the state‐dependent compression factors per service‐class 
, which have a similar role to 
and have already been described in the E‐EMLM. The only difference is that apart from 
, which are given by (3.8), we should also define 
to account for retry calls of service‐class 
in state 
. The form of 
is the following [1]:
where 
, 
, and
4.1.2 The Analytical Model
4.1.2.1 Steady State Probabilities
To describe the analytical model in the steady state, we consider a link of capacity 
b.u. that accommodates calls of two service‐classes with traffic parameters: (
) for service‐class 1 and (
) for service‐class 2. Calls of service‐class 2 have retry parameters with 
and 
. Let 
be the virtual capacity so that the maximum permitted bandwidth compression is 
for calls of both service‐classes.
Although the E‐SRM is a non‐PFS model, we will use the LB of (3.11), initially for calls of service‐class 1:
where 
with 
, and
Based on (4.4) and multiplying both sides of (4.3) with 
, we have:
where 
and the values of 
are given by (4.2).
Based on the CAC of the E‐SRM, we consider the following LB equations for calls of service‐class 2:
- (a) No bandwidth compression
(4.6)
where
with 
and(4.7)
Based on (4.7) and multiplying both sides of (4.6) with
, we have:(4.8)
where
and the values of 
are given by ( 4.2). - (b) Bandwidth compression
(4.9)
where
and(4.10)
Based on (4.10) and multiplying both sides of (4.9) with
, we have:(4.11)
where
and the values of 
are given by ( 4.2).
Equations (4.5), (4.8), and (4.11) lead to the following system of equations:
Equations (4.12)–(4.14) can be combined into one equation by assuming that calls with 
are negligible when 
and calls with 
are negligible when 
:
where 
for 
, otherwise 
and 
for 
, otherwise 
.
Note that the approximations introduced in (4.15) are similar to those introduced in the STM of [2].
Since 
, when 
, it is proved in [ 2] (see also (2.9)) that:
for 
and 
for 
, otherwise 
.
Reminder: To prove (4.16), the migration approximation is needed, which assumes that the population of retry calls of service‐class 2 is negligible in states 
.
When 
and based on ( 4.2), ( 4.15) can be written as:
To introduce the link occupancy distribution q(j) in (4.17), we sum both sides of ( 4.17) over the set of states 
:
Since by definition 
, (4.18) is written as:
where 
for 
.
The combination of ( 4.16) and (4.19) gives the following approximate recursive formula for the calculation of 
in the case of two service‐classes, when only calls of service‐class 2 have retry parameters (for 
):
where 
for 
, otherwise 
, and 
for 
, otherwise 
.
In the case of 
service‐classes and assuming that all service‐classes may have retry parameters (4.20) takes the general form:
where 
, 
.
4.1.2.2 CBP, Utilization, and Mean Number of In‐service Calls
Having determined the unnormalized values of 
, we can calculate [1]:
- The CBP of service‐class k calls with
b.u., 
, via:
(4.22)where
is the normalization constant and 
. - The CBP of service‐class
calls with 
b.u., 
, when 
, via:
(4.23)
Note that if
, then 
refers to the 
and the summation in (4.23) should start from 
. - The conditional CBP of service‐class
retry calls given that they have been blocked with their initial bandwidth requirement 
, via:
(4.24)
- The link utilization,
, by (3.23). - The mean number of service‐class k calls with
b.u. in state 
, via:
(4.25)
- The mean number of service‐class
calls with 
b.u. in state 
, via:
(4.26)where
for 
, otherwise 
and 
for 
, otherwise 
. - The mean number of in‐service calls of service‐class
accepted with 
, via:
(4.27)
- The mean number of in‐service calls of service‐class
accepted with 
, via:
(4.28)
4.2 The Elastic Single‐Retry Model under the BR Policy
4.2.1 The Service System
We now consider the E‐SRM under the BR policy (E‐SRM/BR) with BR parameter 
for service‐class 
calls (
). For CAC in the E‐SRM/BR, we consider the following cases:
- (a) If
, no bandwidth compression takes place and the call is accepted in the link with 
b.u. - (b)
If
, then the call is blocked with 
and retries immediately to be connected in the link with 
. Now if:
- b1)
, no bandwidth compression occurs and the retry call is accepted in the system with 
and 
, so that 
, - b2)
, the retry call is blocked and lost, and - b3)
, the retry call is accepted in the system by compressing its bandwidth requirement 
together with the bandwidth of all in‐service calls of all service‐classes. In that case, the compressed bandwidth of the retry call becomes 
where 
is the compression factor, common to all service‐classes. Similarly, all in‐service calls, which have been accepted in the link with 
(or 
), compress their bandwidth to 
(or 
) for 
. After the compression of all calls the link state is 
. The minimum value of the compression factor is 
.
- b1)
As far as the values of 
, 
, and 
are concerned they are determined by (3.8), (4.1), and ( 4.2), respectively.
4.2.2 The Analytical Model
4.2.2.1 Link Occupancy Distribution
In the E‐SRM/BR, the recursive calculation of 
is based on the Roberts method (see Section 1.3.2.2), which leads to the formula [4]:
where 
and 
.
4.2.2.2 CBP, Utilization, and Mean Number of In‐service Calls
Based on (4.29), the following performance measures can be calculated:
- The CBP of service‐class
calls with 
b.u., 
, via:
(4.30)where
is the normalization constant and 
. - The CBP of service‐class
calls with 
b.u., 
, when 
, via:
(4.31)
Note that if
, then 
refers to the 
and the summation in (4.31) should start from 
. - The conditional CBP of service‐class
retry calls given that they have been blocked with their initial bandwidth requirement 
, via:
(4.32)
- The link utilization,
, via (3.23). - The mean number of service‐class
calls with 
b.u. in state 
, via (4.25), and the mean number of service‐class 
calls with 
b.u. in state 
, 
, via (4.26). - The mean number of in‐service calls of service‐class
accepted in the system with 
, via (4.27), and the mean number of in‐service calls of service‐class 
accepted in the system with 
, via (4.28).
4.3 The Elastic Multi‐Retry Model
4.3.1 The Service System
Similar to the MRM, in the elastic multi‐retry model (E‐MRM) a blocked call of service‐class 
can have more than one retry parameter 
for 
, where 
and 
.
To simply describe the CAC, we assume that a service‐class 
call has a peak‐bandwidth requirement of 
b.u. and may retry twice to be connected in the system, the first time with 
and the second time (if blocked with 
) with 
. Then, for call admission, we consider the following cases:
- (a) If
, no bandwidth compression takes place and the call is accepted in the link with 
b.u. - (b) If
, then the call is blocked with 
and retries immediately to be connected in the link with 
. If 
, the retry call is accepted in the system with 
and 
(no bandwidth compression occurs). - (c)
If
, the retry call is blocked with 
and immediately retries with 
. Now if:
- c1)
, the retry call is accepted in the system with 
and 
(no bandwidth compression occurs). - c2)
, the retry call is blocked and lost, and - c3)
, the retry call is accepted in the system by compressing its bandwidth requirement 
together with the bandwidth of all in‐service calls of all service‐classes. In that case, the compressed bandwidth of the retry call becomes 
, where 
is the compression factor, common to all service‐classes. Similarly, all in‐service calls, which have been accepted in the link with 
(or 
or 
), compress their bandwidth to 
(or 
or 
) for 
. After the compression of all calls the link state is 
. The minimum value of the compression factor is 
.
- c1)
Similar to the E‐SRM, when a service‐class 
call, with bandwidth 
(or 
or 
), departs from the system, the remaining in‐service calls of each service‐class 
expand their bandwidth in proportion to their initially assigned bandwidth 
(or 
or 
). After bandwidth compression/expansion, elastic service‐class calls increase/decrease their service time so that the product service time by bandwidth remains constant.
Similar to the E‐SRM, the steady state probabilities in the E‐MRM do not have a PFS. Thus, the unnormalized values of 
can be determined by an approximate but recursive formula, as presented in Section 4.3.2.
To facilitate the recursive calculation of 
, we replace 
by the state‐dependent compression factors per service‐class 
, 
, and 
, 
. The values of 
are given by (3.8), while those of 
are determined by:
where 
, 
, 
, and
4.3.2 The Analytical Model
4.3.2.1 Steady State Probabilities
Following the analysis of Section 4.1.2.1, the calculation of the unnormalized values of 
is based on an approximate but recursive formula whose proof is similar to that of ( 4.21) [ 1]:
where 
, 
and 
.
4.3.2.2 CBP, Utilization, and Mean Number of In‐service Calls
Having determined the unnormalized values of 
via (4.35) we can calculate [ 1]:
- The final CBP of service‐class
calls with their last bandwidth requirement 
b.u., 
, via:
(4.36)where
is the normalization constant. - The CBP of service‐class k calls with
b.u., 
, via ( 4.23). - The conditional CBP of service‐class
retry calls with 
given that they have been blocked with their initial bandwidth requirement 
, via:
(4.37)
- The link utilization,
, by (3.23). - The mean number of service‐class
calls with 
b.u. in state 
, via ( 4.25). - The mean number of service‐class
calls with 
b.u. in state 
, via:
(4.38)
- The mean number of in‐service calls of service‐class
accepted with 
, via ( 4.27). - The mean number of in‐service calls of service‐class
accepted with 
, via:
(4.39)
4.4 The Elastic Multi‐Retry Model under the BR Policy
4.4.1 The Service System
Compared to the E‐SRM/BR, in the elastic multi‐retry model under the BR policy (E‐MRM/BR) with BR parameter 
for service‐class 
calls (
), blocked calls of service‐class 
can retry more than once to be connected in the system.
To facilitate the recursive calculation of 
in the E‐MRM/BR, we replace 
by the state‐dependent compression factors per service‐class 
, and 
, 
. The values of 
and 
are given by (3.8) and (4.33), respectively.
4.4.2 The Analytical Model
4.4.2.1 Steady State Probabilities
Following the analysis of Section 4.2.2.1, the calculation of the unnormalized values of 
is based on an approximate but recursive formula whose proof is similar to that of ( 4.29) [ 4]:
where 
, 
, 
, 
4.4.2.2 CBP, Utilization, and Mean Number of In‐service Calls
Having determined the unnormalized values of 
via (4.40) we can calculate:
- The final CBP of service‐class
calls with their last bandwidth requirement 
b.u., 
, via:
(4.41)where
is the normalization constant. - The CBP of service‐class
calls with 
b.u., 
, via ( 4.31). - The conditional CBP of service‐class
retry calls with 
given that they have been blocked with their initial bandwidth requirement 
, via:
(4.42)
- The link utilization,
, by (3.23). - The mean number of service‐class
calls with 
b.u. in state 
, via ( 4.25). - The mean number of service‐class
calls with 
b.u. in state 
, via (4.38). - The mean number of in‐service calls of service‐class
accepted with 
, via ( 4.27). - The mean number of in‐service calls of service‐class
accepted with 
, via (4.39).
4.5 The Elastic Adaptive Single‐Retry Model
4.5.1 The Service System
In the elastic adaptive single‐retry model (EA‐SRM), we consider a link of capacity 
b.u. that accommodates 
service‐classes which are distinguished into 
elastic service‐classes and 
adaptive service‐classes. Calls of service‐class 
follow a Poisson process with an arrival rate 
and have a peak‐bandwidth requirement of 
b.u. and an exponentially distributed service time with mean 
. The bandwidth compression/expansion mechanism and the CAC of the EA‐SRM are the same as those of the E‐SRM (Section 4.1.1). The only difference is that adaptive calls do not alter their service time after bandwidth compression/expansion.
Similar to the E‐SRM, the steady state probabilities in the EA‐SRM do not have a PFS, since LB is destroyed between adjacent states (see Figure 4.9). Thus, the unnormalized values of 
can be determined by an approximate but recursive formula, as presented in Section 4.5.2.
Figure 4.9 The state space 
and the state transition diagram (Example 4.13).
To facilitate the recursive calculation of 
, we replace 
by the state‐dependent compression factors per service‐class 
and 
which have already been described in the E‐SRM. The only difference compared to the E‐SRM has to do with the determination of 
which is now given by [5]:
where 
and 
.
4.5.2 The Analytical Model
4.5.2.1 Steady State Probabilities
To describe the analytical model in the steady state, we consider a link of capacity 
b.u. that accommodates calls of two service‐classes with traffic parameters: (
) for service‐class 1 and (
) for service‐class 2. Service‐class 1 is adaptive while service‐class 2 is elastic. Only calls of service‐class 2 have retry parameters with 
and 
. Let 
be the limit up to which bandwidth compression is permitted for calls of both service‐classes.
Although the EA‐SRM is a non‐PFS model we will use the LB of ( 4.3), initially for calls of service‐class 1. As far as 
is concerned it is determined by ( 4.4). Based on ( 4.4) and multiplying both sides of ( 4.3) with 
and 
, we have:
where 
and the values of 
are given by ( 4.43).
Based on the CAC of the EA‐SRM, we consider the following LB equations for calls of service‐class 2:
- No bandwidth compression: in this case, we use ( 4.8) of the E‐SRM.
- Bandwidth compression: in this case, we use ( 4.11) of the E‐SRM.
Equations (4.44), ( 4.8) and ( 4.11) lead to the following system of equations:
Equations (4.45)–(4.47) can be combined into one equation by assuming that calls with 
are negligible when 
and calls with 
are negligible when 
:
where 
for 
, otherwise 
and 
for 
, otherwise 
.
At this point, we derive a formula for 
(which is a simplified version of ( 4.43)) by making the following assumptions:
- When
, the bandwidth of all in‐service calls should be compressed by 
, so that:
(4.49)
- We keep the product service time by bandwidth of service‐class
calls (elastic or adaptive) in state 
of the initial Markov chain (with 
) equal to the corresponding product in the same state 
of the modified Markov chain (with 
and 
):
(4.50)
By substituting (4.50) in (4.49) we obtain:
where 
and 
are given by (3.8), and 
by ( 4.1).
Equation (4.51), due to (3.8) and ( 4.1), is written as:
Based on (4.52), we consider again (4.48). Since 
, when 
, we have ( 4.16).
When 
and based on ( 4.52), ( 4.48) can be written as:
since 
, when 
.
To introduce the link occupancy distribution 
in (4.53), we sum both sides of ( 4.53) over the set of states 
:
Since by definition 
, (4.54) is written as:
where 
for 
.
The combination of ( 4.16) and (4.55) gives the following approximate recursive formula for the calculation of 
in the case of two service‐classes when service‐class 1 is adaptive, service‐class 2 is elastic, and only calls of service‐class 2 have retry parameters:
where 
, and 
for 
, otherwise 
, while 
for 
, otherwise 
.
In the case of 
service‐classes and assuming that all service‐classes may have retry parameters, (4.56) takes the general form [ 5]:
where 
.
4.5.2.2 CBP, Utilization, and Mean Number of In‐service Calls
Having determined the unnormalized values of 
, we can calculate:
- The CBP of service‐class
calls with 
b.u., 
, via (4.22). - The CBP of service‐class
calls with 
b.u., 
, via ( 4.23). - The conditional CBP of service‐class
retry calls given that they have been blocked with their initial bandwidth requirement 
, via (4.24). - The link utilization,
, via (3.23). - The mean number of elastic service‐class
calls with 
b.u. in state 
, via ( 4.25). - The mean number of elastic service‐class
calls with 
b.u. in state 
, via ( 4.26). - The mean number of adaptive service‐class
calls with 
b.u. in state 
, via:
(4.58)
- The mean number of adaptive service‐class
calls with 
b.u. in state 
, via:
(4.59)
- The mean number of in‐service calls of service‐class
accepted with 
, via ( 4.27). - The mean number of in‐service calls of service‐class
accepted with 
, via ( 4.28).
4.6 The Elastic Adaptive Single‐Retry Model under the BR Policy
4.6.1 The Service System
We now consider the EA‐SRM under the BR policy (EA‐SRM/BR) with BR parameter 
for service‐class 
calls (
). The CAC in the EA‐SRM/BR is the same as that of the E‐SRM/BR. As far as the values of 
, 
, and 
are concerned they are determined by (3.8), ( 4.1), and ( 4.43), respectively.
4.6.2 The Analytical Model
4.6.2.1 Link Occupancy Distribution
In the EA‐SRM/BR, the recursive calculation of 
is based on the Roberts method (see Section 1.3.2.2), which leads to the formula [ 4]:
where 
.
4.6.2.2 CBP, Utilization, and Mean Number of In‐service Calls
Based on (4.60), the following performance measures can be calculated:
- The CBP of service‐class
calls with 
b.u., 
, via (4.30). - The CBP of service‐class
calls with 
b.u., 
, via ( 4.31). - The conditional CBP of service‐class
retry calls given that they have been blocked with their initial bandwidth requirement 
via (4.32). - The link utilization,
, via (3.23). - The mean number of service‐class
calls with 
b.u. in state 
, via (4.58), and the mean number of service‐class 
calls with 
b.u. in state 
, via (4.59). - The mean number of in‐service calls of service‐class
accepted in the system with 
, via ( 4.27), and the mean number of in‐service calls of service‐class 
accepted in the system with 
, via ( 4.28).
4.7 The Elastic Adaptive Multi‐Retry Model
4.7.1 The Service System
Similar to the E‐MRM, in the elastic adaptive multi‐retry model (EA‐MRM) a blocked call of service‐class 
can have more than one retry parameter 
for 
, where 
and 
. The call admission in the EA‐MRM is the same as the E‐MRM with the exception that adaptive calls do not alter their service time when their bandwidth is compressed/expanded.
Similar to the EA‐SRM, the steady state probabilities in the EA‐MRM do not have a PFS. Thus, the unnormalized values of 
can be determined by an approximate but recursive formula, as presented in Section 4.7.2.
To facilitate the recursive calculation of 
, we replace 
by the state‐dependent compression factors per service‐class 
and 
, 
whose values are given by (3.8) and ( 4.33), respectively. Due to the existence of adaptive traffic, the values of 
are given by the following formula:
where 
, 
and 
.
4.7.2 The Analytical Model
4.7.2.1 Steady State Probabilities
Following the analysis of Section 4.5.2.1, the calculation of the unnormalized values of 
is based on an approximate but recursive formula whose proof is similar to that of ( 4.57) [ 5]:
where 
, and 
.
4.7.2.2 CBP, Utilization, and Mean Number of In‐service Calls
Having determined the unnormalized values of 
via (4.62) we can calculate [ 5]:
- The final CBP of service‐class k calls with their last bandwidth requirement
b.u., 
, via (4.36). - The CBP of service‐class
calls with 
b.u., 
, via ( 4.23). - The conditional CBP of service‐class
retry calls with 
given that they have been blocked with their initial bandwidth requirement 
, via (4.37). - The link utilization,
, by (3.23). - The mean number of elastic service‐class
calls with 
b.u. in state 
, via ( 4.25). - The mean number of elastic service‐class
calls with 
b.u. in state 
, via ( 4.38). - The mean number of adaptive service‐class
calls with 
b.u. in state 
, via ( 4.58). - The mean number of adaptive service‐class
calls with 
b.u. in state 
:
(4.63)
- The mean number of in‐service calls of service‐class
accepted with 
, via ( 4.27). - The mean number of in‐service calls of service‐class
accepted with 
, via ( 4.39).
4.8 The Elastic Adaptive Multi‐Retry Model under the BR Policy
4.8.1 The Service System
Compared to the EA‐SRM/BR, in the elastic adaptive multi‐retry model under the BR policy (EA‐MRM/BR) with BR parameter 
for service‐class 
calls (
), blocked calls of service‐class 
can retry more than once to be connected in the system.
To facilitate the recursive calculation of 
in the EA‐MRM/BR, we replace 
by the state‐dependent compression factors per service‐class 
and 
, 
. The values of 
and 
are given by (3.8) and ( 4.33), respectively.
4.8.2 The Analytical Model
4.8.2.1 Steady State Probabilities
Following the analysis of Section 4.6.2.1, the calculation of the unnormalized values of 
is based on an approximate but recursive formula whose proof is similar to that of ( 4.60) [3]:
where 
4.8.2.2 CBP, Utilization, and Mean Number of In‐service Calls
Having determined the unnormalized values of 
via (4.64) we can calculate:
- The final CBP of service‐class
calls with their last bandwidth requirement 
b.u., 
, via (4.41). - The CBP of service‐class k calls with
b.u., 
, via ( 4.31). - The conditional CBP of service‐class
retry calls with 
given that they have been blocked with their initial bandwidth requirement 
, via (4.42). - The link utilization,
, by (3.23). - The mean number of elastic service‐class
calls with 
b.u. in state 
, via ( 4.25). - The mean number of elastic service‐class
calls with 
b.u. in state 
, via ( 4.38). - The mean number of adaptive service‐class
calls with 
b.u. in state 
, via ( 4.58). - The mean number of adaptive service‐class
calls with 
b.u. in state 
, via (4.63). - The mean number of in‐service calls of service‐class
accepted in the system with 
, via ( 4.27). - The mean number of in‐service calls of service‐class
accepted in the system with 
, via ( 4.39).
4.9 Applications
Since the multirate elastic adaptive retry loss models are a combination of the retry loss models (see Chapter 2) and the elastic adaptive loss models (see Chapter ), the interested reader may refer to Sections 2.11 and 3.7 for possible applications.
4.10 Further Reading
Similar to the previous section, the interested reader may refer to the corresponding sections of Chapter 2(Section 2.12) and Chapter 3(Section 3.8). In addition to these sections, interesting extensions of the models presented in this chapter have been proposed in [6]. More precisely, in [ 6] the case of finite sources is considered as well as the application of the BR and TH policies.
References
- 1 I. Moscholios, V. Vassilakis, J. Vardakas and M. Logothetis, Retry loss models supporting elastic traffic. Advances in Electronics and Telecommunications, Poznan University of Technology, Poland, 2,(3):8–13, September 2011.
- 2 J. Kaufman, Blocking with retrials in a completely shared resource environment. Performance Evaluation, 15(2):99–113, June 1992.
- 3 I. Moscholios, V. Vassilakis, M. Logothetis and J. Vardakas, Erlang–Engset multirate retry loss models for elastic and adaptive traffic under the bandwidth reservation policy. International Journal on Advances in Networks and Services, 7(1&2):12–24, July 2014.
- 4 I. Moscholios, V. Vassilakis, M. Logothetis and M. Koukias, QoS equalization in a multirate loss model of elastic and adaptive traffic with retrials. 5th International Conference on Emerging Network Intelligence, EMERGING, Porto, Portugal, October 2013.
- 5 I. Moscholios, V. Vassilakis, J. Vardakas and M. Logothetis, Call blocking probabilities of elastic and adaptive traffic with retrials. Proceedings of the 8th Advanced International Conference on Telecommunications, AICT, Stuttgart, Germany, 27 May–1 June 2012.
- 6 I. Moscholios, M. Logothetis, J. Vardakas and A. Boucouvalas, Congestion probabilities of elastic and adaptive calls in Erlang–Engset multirate loss models under the threshold and bandwidth reservation policies. Computer Networks, 92(Part 1):1–23, December 2015.