11
Batched Poisson Multirate Elastic Adaptive Loss Models
In this chapter we consider multirate loss models of batched Poisson arriving calls with fixed bandwidth requirements and elastic bandwidth allocation during service. As we have reached the last chapter of the book, the reader should be able to understand the title of this chapter. As a reminder, in the batched Poisson process (in which now a batch consists of two types of calls, elastic or adaptive), simultaneous call‐arrivals (batches) occur at time‐points following a negative exponential distribution. Elastic calls can reduce their bandwidth and simultaneously increase their service time. Adaptive calls can tolerate bandwidth compression without altering their service time.
11.1 The Elastic Erlang Multirate Loss Model with Batched Poisson Arrivals
11.1.1 The Service System
In the elastic Erlang multirate loss model with batched Poisson arrivals (BP‐E‐EMLM), we consider a link of capacity 
b.u. that accommodates elastic calls of 
different service‐classes. Calls of each service‐class 
have the following attributes: (i) they arrive in the link according to a batched Poisson process with rate 
and batch size distribution 
, where 
denotes the probability that there are 
calls in an arriving batch of service‐class 
, (ii) they follow the partial batch blocking discipline (see Chapter 10), and (iii) they request 
b.u. (peak‐bandwidth requirement). To introduce bandwidth compression in the model, the occupied link bandwidth 
can virtually exceed 
up to a limit of 
b.u. Then the call admission is identical to the one presented in the case of the E‐EMLM (see Section 3.1.1).
Similarly, in terms of the system state‐space 
, the CAC is expressed as in the E‐EMLM (see Section 3.1.1). Hence, the TC probabilities of service‐class 
are determined by the state space 
:
where 
, 
is the number of in‐service calls of service‐class 
in the steady state, 
, 
is the steady state probability, and 
.
11.1.2 The Analytical Model
11.1.2.1 Steady State Probabilities
The compression/expansion of bandwidth results in a non PFS for the values of 
, a fact that makes (11.1) inefficient. In order to analyze the service system and provide an approximate but recursive formula for the calculation of the link occupancy distribution, we assume that local flow balance (as described in Chapter 10) does exist across certain levels that separate two adjacent states of 
. To this end, we present the following notations [1]:
The first step in the analysis is to define the levels across which local flow balance will hold. For the state 
, with 
, the level 
separates 
from 
. Then, the “upward” (due to a new service‐class 
batch arrival) probability flow across level 
is given by [ 1]:
where 
, with 
, and 
is the corresponding steady state probability.
Equation (11.2) can be also written as follows:
where 
is the complementary batch size distribution.
The “downward” (due to a service‐class 
call departure) probability flow across level 
is given by [ 1]:
where 
, defined according to (3.2), is the common state dependent factor used to compress/expand the service rate of calls, when their bandwidth is compressed/expanded, since the target is to keep constant the product service time by bandwidth per call.
Due to the compression/expansion mechanism of calls of service‐class 
the local flow balance across level 
is destroyed, i.e., 
or
Similar to the E‐EMLM, 
are replaced by the state‐dependent compression factors per service‐class 
and 
which are defined according to (3.8) and (3.9), respectively.
Having defined 
and 
, we make the assumption (approximation) that the following equation holds: 
or
Before we proceed with the derivation of a recursive formula for the determination of 
, note that the GB equation for state 
is still given by (10.10). In (10.10), the “upward” probability flows are given by (11.3) and the “downward” probability flows are given by (11.4) (if we refer to 
) or by the RHS side of (11.6) (if we refer to 
).
Consider now two different sets of macro‐states: (i) 
and (ii) 
. For the first set, no bandwidth compression takes place (i.e., 
for all 
and 
can be determined by (10.16) [2]). For the second set, ( 11.6) based on (3.8) is written as:
where 
is the offered traffic‐load of service‐class 
calls (in erl).
Multiplying both sides of (11.7) by 
and summing over 
, we have:
Due to (3.9), (11.8) is written as:
Let 
be the state space where exactly 
b.u. are occupied. Then, summing both sides of (11.9) over 
and since 
, we have:
Interchanging the order of summations in (11.10), we have:
since 
.
The combination of (10.16) with (11.11) gives the approximate recursive formula of 
, when 
[ 1]:
where 
is the largest integer less than or equal to 
and 
is the complementary batch size distribution given by 
.
If calls of service k arrive in batches of size 
where 
is given by the geometric distribution with parameter 
(for all k) then (11.12) takes the form:
If 
for 
and 
for 
, for all service‐classes, then we have a Poisson process and the E‐EMLM results (i.e., the recursion of (3.21)).
11.1.2.2 TC Probabilities, CBP, Utilization, and Mean Number of In‐service Calls
The following performance measures can be determined based on ( 11.12):
- The TC probabilities of service‐class
, via the formula:
(11.14)where
is the normalization constant. - The CBP (or CC probabilities) of service‐class
, via the formula [ 1]:
(11.15)
where
denotes the probability that there are 
calls in an arriving batch of service‐class 
and 
is the maximum integer not exceeding 
.If the batch size is geometrically distributed then (11.15) takes the form:
(11.16)
The proof of (11.16) is similar to the proof of (10) in [3] (which gives the CC probability in the BP‐EMLM where the batch size is geometrically distributed) and therefore is omitted.
- The average number of in‐service calls of service‐class
, via (3.24), where the values of 
can be determined via:
(11.17)where
and 
if 
. The proof of (11.17) is similar to that of (3.25) (see [4]) and thus is omitted. - The link utilization, U, via (3.23).
11.2 The Elastic Erlang Multirate Loss Model with Batched Poisson Arrivals under the BR Policy
11.2.1 The Service System
In the elastic Erlang multirate loss model with batched Poisson arrivals under the BR policy (BP‐E‐EMLM/BR) a new service‐class 
call is accepted in the link if, after its acceptance, the link bandwidth 
, where 
is the BR parameter of service‐class 
.
In terms of the system state‐space 
and due to the partial batch blocking discipline, the CAC is identical to that of the E‐EMLM/BR (see Section 3.2.1). As far as the TC probabilities of service‐class 
are concerned, they are determined by ( 11.1), where 
.
The compression factor 
and the state‐dependent compression factor per service‐class 
are defined according to (3.2) and (3.8), respectively.
11.2.2 The Analytical Model
11.2.2.1 Link Occupancy Distribution
In the BP‐E‐EMLM/BR, the unnormalized values of the link occupancy distribution, 
, can be calculated in an approximate way according to the Roberts method (see Section 1.3.2.2).
Based on Roberts method, ( 11.12) takes the form [ 1]:
where 
, 
is the largest integer less than or equal to 
, 
is the complementary batch size distribution given by 
, and 
is determined via (3.34b).
If calls of service 
arrive in batches of size 
where 
is given by the geometric distribution with parameter 
(for all 
) then (11.18) takes the form:
11.2.2.2 TC Probabilities, CBP, Utilization, and Mean Number of In‐service Calls
The following performance measures can be determined based on ( 11.18):
- The TC probabilities of service‐class
, via the formula [ 1]:
(11.20)where
is the normalization constant. - The CBP (or CC probabilities) of service‐class
, via the formula:
(11.21)
If the batch size is geometrically distributed then (11.21) takes the form:
(11.22)
- The average number of in‐service calls of service‐class
, via (3.24), where the values of 
can be determined via ( 11.17) assuming that 
when 
. - The link utilization,
, via (3.23).
11.3 The Elastic Adaptive Erlang Multirate Loss Model with Batched Poisson Arrivals
11.3.1 The Service System
In the elastic adaptive Erlang multirate loss model with batched Poisson arrivals (BP‐EA‐EMLM) we consider a link of capacity 
b.u. that accommodates 
service‐classes which are distinguished into 
elastic service‐classes and 
adaptive service‐classes: 
. The call arrival process remains batched Poisson.
The bandwidth compression/expansion mechanism and the CAC of the BP‐EA‐EMLM are identical to those presented in the case of the E‐EMLM (Section 3.1.1). The only difference is in (3.3), which is applied only on elastic calls (adaptive calls tolerate bandwidth compression without altering their service time). As far as the TC probabilities of service‐class 
calls are concerned they can be determined via ( 11.1).
11.3.2 The Analytical Model
11.3.2.1 Steady State Probabilities
Due to the compression/expansion mechanism of calls of service‐class 
the local flow balance across the level 
is destroyed, i.e., (11.5) holds for elastic calls while the corresponding formula for adaptive calls is 5,6:
Similar to the BP‐E‐EMLM, 
are replaced by the state‐dependent compression factors per service‐class 
, 
and 
, which are defined according to (3.8) and (3.47), respectively.
Having defined 
and 
, we make the assumption (approximation) that the following equations hold: 
, or
where the only difference between (11.24) and (11.25) is in 
.
Before we proceed with the derivation of a recursive formula for the determination of 
, note that the GB equation for state 
is given by (10.10). In (10.10), the “upward” probability flows are given by ( 11.3) and the “downward” probability flows are given by (i) ( 11.5) and (11.23) for elastic and adaptive calls, respectively (if we refer to 
), or (ii) the RHS of ( 11.24), ( 11.25) (if we refer to 
).
Similar to the BP‐E‐EMLM, we consider two different sets of macro‐states: (i) 
and (ii) 
. For the first set, no bandwidth compression takes place and 
can be determined by (10.16) [ 2]. For the second set, we substitute (3.8) in ( 11.24) and ( 11.25) to have:
where 
is the offered traffic‐load of service‐class 
calls (in erl).
Multiplying both sides of (11.26a) by 
and summing over 
, we have:
Similarly, multiplying both sides of (11.26b) by 
and 
, and summing over 
, we have:
By adding (11.27) and (11.28), we have:
Based on (3.47), (11.29) is written as:
Summing both sides of (11.30) over 
and based on the fact that 
, we have:
By interchanging the order of summations in (11.31), we obtain:
or
because 
.
The combination of (10.16) with (11.33) gives the approximate recursive formula of 
, when 
[ 5, 6]:
where 
, 
is the largest integer less than or equal to 
and 
is the complementary batch size distribution given by 
.
If calls of service 
arrive in batches of size 
where 
is given by the geometric distribution with parameter 
(for all 
) then (11.34) takes the form:
If 
for 
and 
for 
, for all service‐classes, then we have a Poisson process and the EA‐EMLM results (i.e., the recursion of (3.57)).
11.3.2.2 TC Probabilities, CBP, Utilization, and Mean Number of In‐service Calls
The following performance measures can be determined based on ( 11.34):
- The TC probabilities of service‐class
, via ( 11.14). - The CBP (or CC probabilities) of service‐class
, via ( 11.15) or via ( 11.16) if the batch size is geometrically distributed. - The average number of in‐service calls of service‐class
, via (3.24), where the values of 
can be determined via (11.36) for elastic traffic and via (11.37) for adaptive traffic:
(11.36)
(11.37)where
and 
if 
. - The link utilization,
, via (3.23).
11.4 The Elastic Adaptive Erlang Multirate Loss Model with Batched Poisson Arrivals under the BR Policy
11.4.1 The Service System
In the elastic adaptive Erlang multirate loss model with batched Poisson arrivals under the BR policy (BP‐EA‐EMLM/BR) a new service‐class 
call is accepted in the link if, after its acceptance, the link bandwidth 
, where 
is the BR parameter of service‐class 
.
In terms of the system state‐space 
and due to the partial batch blocking discipline, the CAC is identical to that of the E‐EMLM/BR (see Section 3.2.1). The only difference is in (3.3), which is applied only on elastic calls. As far as the TC probabilities of service‐class 
are concerned, they are given by ( 11.1), where 
.
The compression factor 
and the state‐dependent compression factor per service‐class 
, 
are defined according to (3.2) and (3.8), respectively, while the values of 
are given by (3.47).
11.4.2 The Analytical Model
11.4.2.1 Link Occupancy Distribution
In the BP‐EA‐EMLM/BR, the unnormalized values of the link occupancy distribution, 
, can be calculated in an approximate way according to the Roberts method (see Section 1.3.2.2).
Based on the Roberts method, ( 11.34) takes the form [ 5]:
where 
= 
, 
is the largest integer less than or equal to 
, 
is the complementary batch size distribution given by 
, and 
is determined via (3.34b).
If calls of service 
arrive in batches of size 
where 
is given by the geometric distribution with parameter 
(for all 
), then (11.38) becomes:
11.4.2.2 TC Probabilities, CBP, Utilization, and Mean Number of In‐service Calls
The following performance measures can be determined based on ( 11.38):
- The TC probabilities of service‐class
, via ( 11.20). - The CBP (or CC probabilities) of service‐class
, via ( 11.21) or via ( 11.22) if the batch size is geometrically distributed. - The average number of in‐service calls of service‐class
, via (3.24), where the values of 
can be determined via ( 11.36) and ( 11.37) assuming that 
when 
. - The link utilization,
, via (3.23).
11.5 Applications
Since the batched Poisson multirate elastic adaptive loss models are a combination of the elastic adaptive multirate loss models (see Chapter 3) and the batched Poisson multirate loss models (see Chapter 10 ), the interested reader may refer to Sections 3.7 and 10.4 for possible applications.
11.6 Further Reading
Similar to the previous section, the interested reader may refer to the corresponding sections of Chapter 3 (Section 3.8) and Chapter 10 (Section 10.5). In addition to these sections, a possible interesting extension could be the recent work of [8]. In [ 8], a self‐optimizing 4G wireless network is considered. The network consists of a group of cells that accommodates elastic and adaptive calls of random or quasi‐random input and incorporates a mechanism for load balancing between the cells of a group. The determination of CBP is based on approximate formulas whose accuracy has been verified via simulation. The case of bursty traffic in such a network could be studied with the aid of the models in this chapter.
References
- 1 I. Moscholios, J. Vardakas, M. Logothetis and A. Boucouvalas, A batched Poisson multirate loss model supporting elastic traffic under the bandwidth reservation policy. Proceedings of the IEEE International Conference on Communications, ICC, Kyoto, Japan, June 2011.
- 2 J. Kaufman and K. Rege, Blocking in a shared resource environment with batched Poisson arrival processes. Performance Evaluation, 24(4):249–263, February 1996.
- 3 E. van Doorn and F. Panken, Blocking probabilities in a loss system with arrivals in geometrically distributed batches and heterogeneous service requirements. IEEE/ACM Transactions on Networking, 1(6):664–667, December 1993.
- 4 S. Racz, B. Gero and G. Fodor, Flow level performance analysis of a multi‐service system supporting elastic and adaptive services. Performance Evaluation, 49(1–4):451–469, September 2002.
- 5 I. Moscholios, J. Vardakas, M. Logothetis and A. Boucouvalas, QoS guarantee in a batched Poisson multirate loss model supporting elastic and adaptive traffic. Proceedings of the IEEE International Conference on Communications, ICC, Ottawa, Canada, June 2012.
- 6 I. Moscholios, J. Vardakas, M. Logothetis and A. Boucouvalas, Congestion probabilities in a batched Poisson multirate loss model supporting elastic and adaptive traffic. Annals of Telecommunications, 68(5):327–344, June 2013.
- 7 I. Moscholios and M. Logothetis, The Erlang multirate loss model with batched Poisson arrival processes under the bandwidth reservation policy. Computer Communications, 33(Supplement 1):S167–S179, November 2010.
- 8 M. Glabowski, S. Hanczewski and M. Stasiak, Modelling load balancing mechanisms in self‐optimizing 4G mobile networks with elastic and adaptive traffic. IEICE Transactions on Communications, E99‐B(8):1718–1726, August 2016.