2
Multirate Retry Threshold Loss Models

We consider multirate loss models of random arriving calls with elastic bandwidth requirements and fixed bandwidth allocation during service. Calls may retry several times upon arrival (requiring less bandwidth each time) in order to be accepted for service. Alternatively, calls may request less bandwidth upon arrival, according to the occupied link bandwidth indicated by threshold(s).

2.1 The Single‐Retry Model

2.1.1 The Service System

In the single‐retry model (SRM), a link of capacity images b.u. accommodates Poisson arriving calls of images service‐classes, under the CS policy. A new call of service‐class images has a peak‐bandwidth requirement of images b.u. and an exponentially distributed service time with mean images. If the initially required b.u. are not available in the link, the call is blocked, but immediately retries to be connected with bandwidth requirement images b.u., while the mean of the new (exponentially distributed) service time increases to images, so that the product bandwidth requirement by service time images remains constant [1,2]. If the images b.u. are not available, the call is blocked and lost (Figure 2.1). The CAC mechanism of a call of service‐class images is depicted in Figure 2.2. A new call of service‐class images is blocked with images b.u. if images and is accepted with images b.u., if images, where images and images are the in‐service calls of service‐class images (in the steady state of the system) accepted with images b.u., respectively. The comparison of the SRM with the EMLM reveals the following basic differences:

  1. (i) The steady state probabilities in the SRM do not have a PFS, since the notion of LB between adjacent states does not hold (see Example 2.1). Because of this, the unnormalized values of images are determined via an approximate, but recursive, formula, as Section 2.1.2 shows.
  2. (ii) In the EMLM, the steady state vector of all in‐service calls of all service‐classes is images, while in the SRM the corresponding vector becomes images. This dimensionality increase means that only quite small problems can be solved exactly (see Example 2.1).
  3. (iii) In the EMLM, the calculation of images via the Kaufman–Roberts recursion (1.39) is insensitive to the service time distribution [3]. In the SRM, this insensitivity property does not hold. However, numerical examples in [ 2] show that the CBP obtained for various service time distributions are quite close.
Schematic illustrating the service system of the SRM displaying a vertical bar having brackets marking the available bandwidth and occupied bandwidth j. Arrows at the top left are labeled bk and bkr.

Figure 2.1 Service system of the SRM.

Flowchart illustrating the CAC mechanism for a new call in the SRM, starting from NEW CALL to j + bk ≤ C, to The call retries, to j + bkr ≤ C leading to The call is blocked & lost and The call leaves the system.

Figure 2.2 The CAC mechanism for a new call in the SRM.

State space Ω (CS policy) and state transition diagram (Example 2.1) displaying circles labeled 0,(1,1); 0,(0,2); 1,(0,2); 0(0,1); 1,(0,1); 2(0,1); 3(0,1); etc. linked by curved arrows with corresponding labels.

Figure 2.3 The state space images (CS policy) and the state transition diagram (Example 2.1).

2.1.2 The Analytical Model

2.1.2.1 Steady State Probabilities

To describe the analytical model in the steady state, let us concentrate on a single link of capacity images b.u. that accommodates two service‐classes with the following traffic characteristics: images. Blocked calls of service‐class 2 can retry with parameters images, while blocked calls of service‐class 1 are not allowed to retry. Although the SRM does not have a PFS, we assume that the LB equation (1.51) proposed in the EMLM does hold, that is:

(2.1)equation

This assumption (approximation) is important for the derivation of an approximate but recursive formula for the calculation of images. The aforementioned equation expresses the fact that no call blocking occurs in state images, if there are available images b.u. images. If images, when a new call of service‐class 2 arrives in the system, then this call is blocked and retries to be connected with images b.u. If images, then, the retry call will be accepted in the system. To describe the latter case, we need an additional LB equation [ 2]:

(2.2)equation

where images is the mean number of service‐class images calls accepted in the system with images in state images.

Dividing (2.2) by images and multiplying by images, we obtain [ 2]:

(2.3)equation

where images is the offered traffic‐load of service‐class 2 calls with images.

Equation (2.1) can be written as images. Then, by multiplying both sides with images and summing up for images, we have:

(2.4)equation

Adding (2.3) to (2.4), and since images, we have:

(2.5)equation

Apart from the assumption of the LB equation ( 2.1), another approximation is necessary for the recursive calculation of images [ 2]:

(2.6)equation

Equation (2.6) expresses the so‐called migration approximation [ 1, 2,4], according to which the number of calls accepted in the system with other than the maximum bandwidth requirement is negligible within a state space, called the migration space. In this space, the value of images is negligible compared to images when images. For service‐class images (with images):

(2.7)equation

Equation ( 2.4) due to ( 2.6) is written as:

(2.8)equation

The combination of (2.5) and (2.8) is achieved through the use of a binary parameter (images), and gives an approximate but recursive formula for the determination of images in the SRM, considering two service‐classes where only calls of service‐class 2 can retry [ 1, 2]:

(2.9)equation

where images, and images when images (otherwise images).

The symbol images is used to distinguish retry models, where only the migration approximation exists, from other models (e.g., thresholds models presented in Section 2.5) where the symbol images is used and additional approximations are considered.

The generalization of (2.9) for images service‐classes, where all service‐classes may retry, is as follows [ 2]:

(2.10)equation

where images and images when images (otherwise images).

Note that the variable images in (2.10) expresses the migration approximation, i.e., (2.7).

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

Having determined the unnormalized values of images, we can calculate the following performance measures:

  • The CBP of service‐class images calls with images b.u. (i.e., the actual CBP of service‐class images with retrial), images, via the following formula [ 2]:
    (2.11)equation
    where images is the normalization constant.
  • The CBP of service‐class images calls with images b.u. (i.e., the actual CBP of service‐class images without retrial, or the retry probability in case of service‐class images with retrial), images, via:
    (2.12)equation
  • The conditional CBP of service‐class images retry calls given that they have been blocked with their initial bandwidth requirement images, via:
    (2.13)equation
  • The link utilization, images, via (1.54).
  • The mean number of service‐class k calls with images b.u. in state images, via:
    (2.14)equation
  • The mean number of service‐class images calls with images b.u. in state images, via:
    (2.15)equation
    where images when images (otherwise images).
  • The mean number of in‐service calls of service‐class k accepted in the system with images, via:
    (2.16)equation
  • The mean number of in‐service calls of service‐class k accepted in the system with images, via:
    (2.17)equation

2.2 The Single‐Retry Model under the BR Policy

2.2.1 The Service System

In the SRM under the BR policy (SRM/BR), images b.u. are reserved to benefit calls of all other service‐classes apart from service‐class images. The application of the BR policy in the SRM is similar to that of the EMLM/BR, as the following example shows.

2.2.2 The Analytical Model

2.2.2.1 Steady State Probabilities

To calculate the link occupancy distribution in the steady state of the SRM/BR, we prefer the Roberts method to the Stasiak–Glabowski method. The latter is more complex compared to the Roberts method and does not provide more accurate results (compared to simulation) in retry loss models and threshold loss models (see Section 2.6, below) [5].

Based on the Roberts method, ( 2.10) takes the form [ 4]:

(2.18)equation

where images, images, when images (otherwise images), images is given by (1.65) and, similarly, images by:

(2.19)equation

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

Based on (2.18) and (2.19), we can calculate the following performance measures:

  • The CBP of service‐class images calls with images b.u., images, via the following formula [ 4]:
    (2.20)equation
    where images is the normalization constant.
  • The CBP of service‐class images calls with images b.u., images (i.e., the actual CBP of service‐class images without retrials, or retry probability in case of service‐class images with retrials), can be determined via (1.66), while the conditional CBP of service‐class images retry calls given that they have been blocked with their initial bandwidth requirement images, via (2.13), while subtracting images from images and from images.
  • The link utilization, images, is given by (1.54).
  • The mean number of service‐class images calls with images b.u. in state images, images, is given by (1.67), while the mean number of service‐class images calls with images b.u. in state images, images, is given by:
    (2.21)equation
    where images when images (otherwise images).
  • The mean number of in‐service calls of service‐class images accepted in the system with images, is calculated by (2.16), while the mean number of in‐service calls of service‐class images accepted in the system with images, by (2.17).

Table 2.1 CBP of Example 2.5 (SRM, images b.u. and images b.u.).

images b.u. images b.u.
CBP Analytical results Simulation results Analytical results Simulation results
images 0.0479 0.0493 images 1.2745e‐4 0.0004 0.0004 images 9.0281e‐6
images 0.2997 0.2986 images 5.1219e‐4 0.0031 0.0029 images 5.6690e‐5
images 0.5089 0.5077 images 4.7558e‐4 0.0063 0.0060 images 6.8768e‐5
images 0.7360 0.7430 images 4.8105e‐4 0.0128 0.0125 images 1.0325e‐4
images 0.5089 0.5081 images 6.3207e‐4 0.0063 0.0060 images 8.0577e‐5
images 0.6914 0.6844 images 1.4349e‐3 0.4955 0.4871 images 4.6728e‐3

Table 2.2 CBP of Example 2.5 (SRM/BR, images b.u. and images b.u.).

images b.u. images b.u.
CBP Analytical results Simulation results Analytical results Simulation results
images 0.3804 0.3825 images 3.3290e‐4 0.00485 0.00485 images 1.1740e‐4
images 0.3804 0.3826 images 4.2058e‐4 0.00485 0.00486 images 1.6266e‐4
images 0.3804 0.3824 images 4.2857e‐4 0.00485 0.00485 images 1.3816e‐4
images 0.3804 0.3825 images 6.0689e‐4 0.00485 0.00487 images 1.0434e‐4
CBP vs. C (b.u.) displaying 9 curves for B1 (EMLM), B1 (SRM), B2 (EMLM), B2 (SRM), B3 (EMLM), B3 (SRM), B4 (EMLM), B4 (SRM), and B4r* (SRM).

Figure 2.5 CBP in the SRM and EMLM, for various values of images (Example 2.5).

Link utilization vs. C (b.u.) displaying 2 ascending curves for SRM (solid) and EMLM (dotted) that are almost coinciding to each other.

Figure 2.6 Link utilization in the SRM and EMLM (Example 2.5).

CBP vs. C (b.u.) displaying 4 descending curves for EMLM/BR (set 1, B4) (solid), EMLM/BR (set 2) (dotted), SRM/BR (dashed), and EMLM/BR (set 1, B1, B2, B3) (dash-dotted).

Figure 2.7 CBP in the SRM/BR and the EMLM/BR for various values of images (Example 2.5).

2.3 The Multi‐Retry Model

2.3.1 The Service System

In the multi‐retry model (MRM), calls of service‐class images can retry not only once, but several times, in order to be accepted in the system [ 1, 2]. Let images be the number of retrials for calls of service‐class images, and assume that images, where images is the required bandwidth of a service‐class images call in the images retry, images. Then a service‐class images call is accepted in the system with images b.u. if images. By definition, images and images.

2.3.2 The Analytical Model

2.3.2.1 Steady State Probabilities

Following the analysis of Section 2.1.2.1, we have to assume in the MRM the existence of both LB and the migration approximation. According to the migration approximation, the mean number of service‐class k calls in state images, images, accepted with images b.u., is negligible when images, where images. This means that service‐class k calls with images are limited in the area images. Based on [1, 2], the unnormalized values of images can be determined by the following recursive formula:

(2.22)equation

where images when images (otherwise images).

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

Having determined the unnormalized values of images, we can determine the following performance measures:

  • The CBP of service‐class images calls (with their last bandwidth requirement images), images, are determined as follows (if images is the normalization constant) [ 2]:
    (2.23)equation
  • The images of service‐class images calls of images b.u., images, is determined via (2.12), while the conditional probability of service‐class images retry calls, requesting images b.u. given that they have been blocked with their initial bandwidth requirement images, is defined as:
    (2.24)equation
  • The link utilization, images, is given by (1.54).
  • The mean number of service‐class images calls with images b.u. in state images, images, is given by (2.14), while the mean number of service‐class images calls with images b.u. in state images, images, is given by:
    (2.25)equation
    where images when images (otherwise images).
  • The mean number of in‐service calls of service‐class k accepted in the system with images, is calculated by ( 2.16), while the mean number of in‐service calls of service‐class images accepted in the system with images is determined by:
    (2.26)equation

2.4 The Multi‐Retry Model under the BR Policy

2.4.1 The Service System

Obviously, in the MRM under the BR policy (MRM/BR), unlike the SRM/BR, blocked calls of service‐class images can retry more than once to be connected in the system.

State space Ω (BR policy) and state transition diagram (Example 2.8) displaying circles labeled 0,(0,1,1); 0,(0,1,0); 0,(1,0,0); 1(0,1,0); 0,(0,0,1); etc. linked by curved arrows with corresponding labels.

Figure 2.9 The state space images (BR policy) and the state transition diagram (Example 2.8).

2.4.2 The Analytical Model

2.4.2.1 Steady State Probabilities

Based on the Roberts method, we calculate the unnormalized link occupancy distribution in the steady state of the MRM/BR by modifying ( 2.22) as follows [ 4]:

(2.27)equation

where images when images (otherwise images), and images is given by (1.65) and, similarly, images by:

(2.28)equation

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

Based on (2.27) and (2.28), we can determine the following performance measures:

  • The CBP of service‐class images calls with images b.u., images, by the following formula [ 4]:
    (2.29)equation
    where images is the normalization constant.
  • The CBP of service‐class images calls with images b.u., images, via (1.66).
  • The conditional CBP of service‐class images retry calls, with images, given that they have been blocked with their initial bandwidth requirement images, via (2.24), while subtracting images from images and from images.
  • The link utilization, images, via (1.54).
  • The mean number of service‐class images calls with images b.u. in state images, images, via (1.67).
  • The mean number of service‐class images calls with images b.u. in state images, images, via:
    (2.30)equation
    where images when images (otherwise images).
  • The mean number of in‐service calls of service‐class images accepted with images, via ( 2.16).
  • The mean number of in‐service calls of service‐class images accepted in the system with images, via (2.26).

2.5 The Single‐Threshold Model

2.5.1 The Service System

In the single‐threshold model (STM), the requested b.u. and the corresponding service time of a new call are related to the value images of the occupied link bandwidth (upon the new call arrival). More precisely, the following CAC is applied. When the value of images is lower or equal to a threshold images, then a new call of service‐class images is accepted in the system with its initial requirements images. Otherwise, if images, the call tries to be connected in the system with images, where images and images, so that the product bandwidth requirement by service time remains constant [ 1]. This means that, contrary to the SRM, a call does not have to be blocked in order to retry with lower bandwidth requirement. If the images b.u. are not available the call is blocked and lost.

The comparison of the STM with the SRM reveals the following basic similarities and differences:

  • The steady state probabilities in the STM do not have a PFS, similar to the SRM (see Example 2.11). Thus, the unnormalized values of images are determined via an approximate, but recursive, formula, as Section 2.5.2 shows.
  • In the STM, the steady state vector of all in‐service calls of all service‐classes is images. Although a similar vector is defined for the SRM, an SRM system and an STM system that accommodate the same service‐classes may have a different number of possible states, depending on the value of the threshold images.
  • The CBP results obtained in the STM are sensitive to the service time distribution [ 1].
  • Setting the value of the threshold images results in the same CBP for both the SRM and STM.

2.5.2 The Analytical Model

2.5.2.1 Steady State Probabilities

Aiming at deriving a recursive formula for the calculation of images (the unnormalized link occupancy distribution), we consider a link of capacity images b.u. that accommodates calls of two service‐classes, whose initial bandwidth requirements are images and images b.u., respectively. Calls of each service‐class arrive in the link according to a Poisson process with means images and images, and have exponentially distributed service times with means images and images, respectively. If images upon the arrival of a service‐class 2 call, then this call requests from the system images and images. No such option is considered for calls of service‐class 1.

Although the STM does not have a PFS, we assume that the LB equation (1.51) does hold for calls of service‐class 1, for images:

(2.31)equation

For calls of service‐class 2, we assume the existence of LB between adjacent states that can be expressed as follows [ 2]:

(2.32)equation
(2.33)equation

where images is the mean number of service‐class 2 calls with images in state images.

Equations (2.31)–(2.33) lead to the following system of equations [ 2]:

(2.34)equation
(2.35)equation
(2.36)equation

For (2.34), the following approximation is adopted: the value of images in state images is negligible when images. This approximation is similar to the migration approximation used in the SRM. For (2.36), the following approximation is applied: the value of images in state images is negligible when images. This approximation is named the upward migration approximation and is different from the migration approximation since it considers negligible the population of calls with their initial bandwidth requirement [ 2]. The error introduced by the upward migration approximation in the calculation of images can be higher than the corresponding error introduced by the migration approximation of the SRM, especially when the offered traffic‐load is light. In that case, it is highly probable that calls are accepted in the system with their initial bandwidth requirement [2].

Based on the above‐mentioned approximations, we have (for images):

(2.37)equation

where images, images, and images.

Note that in (2.37), images expresses the upward migration approximation and images expresses the migration approximation.

In the general case of images service‐classes, the approximate but recursive formula for images is the following [ 2]:

(2.38)equation

where images, images and images.

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

Having determined the unnormalized values of images, we can calculate the following performance measures:

  • The CBP of service‐class images calls with images b.u., images, via the following formula (if images is the normalization constant) [ 2]:
    (2.39)equation
  • The CBP of service‐class k calls with images b.u. (assuming that they have no option for images), images, via ( 2.12).
  • The conditional CBP of service‐class images calls with images given that images, via:
    (2.40)equation

    Note that if images, then (2.40) is identical to ( 2.13) of the SRM.

  • The link utilization, images, via (1.54).
  • The mean number of service‐class images calls with images b.u. in state images, via:
    (2.41)equation
  • The mean number of service‐class images calls with images b.u. in state images, via:
    (2.42)equation
  • The mean number of in‐service calls of service‐class k accepted in the system with images, images, via ( 2.16).
  • The mean number of in‐service calls of service‐class k accepted in the system with images, via:
    (2.43)equation

2.6 The Single‐Threshold Model under the BR Policy

2.6.1 The Service System

In the STM under the BR policy (STM/BR), images b.u. are reserved to benefit calls of all other service‐classes apart from service‐class images. The application of the BR policy in the STM is similar to that of the SRM/BR as the following example shows.

2.6.2 The Analytical Model

2.6.2.1 Steady State Probabilities

Similar to the SRM/BR, we adopt the Roberts method for the calculation of images in the STM/BR.

Based on the Roberts method, ( 2.38) takes the form [ 4]:

(2.44)equation

where images, images, images, images is given by (1.65), and

(2.45)equation

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

Based on (2.44) and (2.45), we can determine the following performance measures:

  • The CBP of service‐class images calls with images b.u., images, via the formula [ 4]:
    (2.46)equation
    where images is the normalization constant.
  • The CBP of service‐class k calls with images b.u., images, via (1.66).
  • The conditional CBP of service‐class k calls with images given that images, images, via ( 2.40), while subtracting the BR parameter images from images.
  • The link utilization, images, via (1.54).
  • The mean number of service‐class k calls with images b.u. in state images, images, via:
    (2.47)equation
    equation
  • The mean number of service‐class k calls with images b.u. in state images, images, via:
    (2.48)equation
    where images.
  • The mean number of in‐service calls of service‐class k accepted in the system with images, via ( 2.16).
  • The mean number of in‐service calls of service‐class k accepted in the system with images, via (2.43).
Left: CBP vs. C with 6 descending curves for B1 (J0 = C - 3b4, STM), B1 (J0 = C - 2b4, STM), B1 (SRM), B2 (SRM), etc. Right: CBP vs. C with 6 descending curves for B3 (J0 = C - 3b4, STM), B3 (J0 = C - 2b4, STM), B3 (SRM), B4r* (SRM), etc.

Figure 2.15 Left: CBP of service‐classes 1, 2 in the STM and SRM versus various values of images. Right: The corresponding graphs for service‐classes 3, 4 (Example 2.15).

Table 2.4 CBP of Example 2.15 (STM, images or images b.u., and images).

images b.u. images b.u.
CBP Analytical results Simulation results Analytical results Simulation results
images 0.0474 0.0479 images 2.9368e‐4 0.0003 0.00026 images 1.6765e‐5
images 0.2953 0.2948 images 3.3213e‐4 0.0025 0.0019 images 3.3650e‐5
images 0.5086 0.5074 images 3.6020e‐4 0.0054 0.0042 images 3.0290e‐5
images 0.5361 0.5302 images 3.2453e‐4 0.1579 0.1363 images 2.3964e‐3

Table 2.5 CBP of Example 2.15 (STM, images or images b.u., and images).

images b.u. images b.u.
CBP Analytical results Simulation results Analytical results Simulation results
images 0.0475 0.0480 images 1.4185e‐4 0.0003 0.00017 images 1.2923e‐5
images 0.2949 0.2948 images 5.6484e‐4 0.0021 0.00137 images 2.9135e‐5
images 0.5082 0.5074 images 6.7936e‐4 0.0047 0.00297 images 6.4709e‐5
images 0.5124 0.5113 images 7.5074e‐4 0.0702 0.0511 images 8.6489e‐4

Table 2.6 CBP of Example 2.15 (STM/BR, images or images b.u., and images).

images b.u. images b.u.
CBP Analytical results Simulation results Analytical results Simulation results
images 0.3783 0.3799 images 5.0062e‐4 0.00410 0.00332 images2.6103e‐5
images 0.3783 0.3799 images 4.6590e‐4 0.00410 0.00333 images4.1082e‐5
images 0.3783 0.3799 images 7.1394e‐4 0.00410 0.00332 images3.8769e‐5
images 0.3783 0.3799 images 6.9529e‐4 0.00410 0.00333 images3.6205e‐5
Equalized CBP vs. C (b.u.) displaying 3 descending curves for SRM/BR (solid), SRM/BR (J0 = C - 2b4) (dotted), and SRM/BR (J0 = C - 3b4) (dashed).

Figure 2.16 CBP in the STM/BR and SRM/BR versus images and two values of images (Example 2.15).

2.7 The Multi‐Threshold Model

2.7.1 The Service System

In the multi‐threshold model (MTM), there exist images different thresholds which are common to all service‐classes [ 1, 2]. A call of service‐class images with initial requirements images can use, depending on the occupied link bandwidth, one of the images requirements images, where the pair images is used when images (where images. The maximum possible threshold is images, while images. As far as the bandwidth requirements of a service‐class images call are concerned, we assume that they decrease as images increases, i.e., images, while by definition images (see Figure 2.17).

Schematic illustrating the MTM principle of operation depicted by a close shape containing a vertical scale with tick marks labeled C, J2, J1, J0, and 0 pointed by arrows labeled bkc3, bkc2, bkc1, and bk = bkc0.

Figure 2.17 The MTM principle of operation.

2.7.2 The Analytical Model

2.7.2.1 Steady State Probabilities

To describe the MTM in steady state, the following LB equations are considered:

(2.49)equation

and

(2.50)equation

where images denotes the mean number of service‐class images calls, with images, in state images.

Similar to the analysis of the STM and based on (2.49) and (2.50), we have the following recursive formula for the calculation of images [ 1, 2]:

(2.51)equation

where images images and images.

Note that in (2.51), images expresses the upward migration approximation, while images expresses the migration approximation.

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

Based on ( 2.51), we can determine the following performance measures:

  • The CBP of service‐class images calls with images b.u., images, via the formula [ 2]:
    (2.52)equation
    where images is the normalization constant.
  • The CBP of service‐class images calls with images b.u., images, via (1.66).
  • The conditional CBP of service‐class images calls with images given that images, via the formula:
    (2.53)equation
  • The link utilization, images, via (1.54).
  • The mean number of service‐class images calls with images b.u. in state images, via:
    (2.54)equation
    where images
  • The mean number of service‐class images calls with images b.u. in state images, via:
    (2.55)equation
    where images.
  • The mean number of in‐service calls of service‐class images accepted in the system with images, via ( 2.16).
  • The mean number of in‐service calls of service‐class images accepted in the system with images, via:
    (2.56)equation

2.8 The Multi‐Threshold Model under the BR Policy

2.8.1 The Service System

In the MTM under the BR policy (MTM/BR), images b.u. are reserved to benefit calls of all other service‐classes apart from service‐class images. The application of the BR policy in the MTM is similar to that of the MRM/BR.

2.8.2 The Analytical Model

2.8.2.1 Steady State Probabilities

Similar to the MRM/BR, we adopt the Roberts method for the calculation of the unnormalized link occupancy distribution images in the MTM/ BR.

Based on the Roberts method, ( 2.51) takes the form [ 4]:

(2.57)equation

where images images,   images,  and

(2.58)equation

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

Based on (2.57) and (2.58), we can determine the following performance measures:

  • The CBP of service‐class k calls with images b.u., images, via the formula [ 4]:
    (2.59)equation
    where images is the normalization constant.
  • The CBP of service‐class images calls with images b.u., images, via (1.66).
  • The conditional CBP of service‐class images calls with images given that images, via the formula:
    (2.60)equation
  • The link utilization, images, via (1.54).
  • The mean number of service‐class images calls with images b.u. in state images, via:
    (2.61)equation
    where images
  • The mean number of service‐class images calls with images b.u. in state images, via:
    (2.62)equation
    where images.
  • The mean number of in‐service calls of service‐class images accepted in the system with images, via ( 2.16).
  • The mean number of in‐service calls of service‐class images accepted in the system with images, via (2.56).
Left: CBP vs. threshold sets with 3 descending curves for (top-bottom) B3 (solid), B2 (dotted), and B1 (dashed). Right: Conditional CBP vs. threshold sets with a descending solid curve for B*4c4.

Figure 2.18 Left: CBP of service‐classes 1, 2, and 3 versus the sets of thresholds. Right: Conditional CBP of service‐class 4 versus the sets of thresholds (MTM) (Example 2.16).

Equalized CBP vs. threshold sets displaying 3 descending curves for C = 300 b.u. (solid, top), C = 310 b.u. (dotted, middle), and C = 320 b.u. (dashed, bottom).

Figure 2.19 Equalized CBP of all service‐classes versus the sets of thresholds (MTM/BR) (Example 2.16).

2.9 The Connection Dependent Threshold Model

2.9.1 The Service System

In the connection dependent threshold model (CDTM), bandwidth and service time requests depend on the total number images of occupied b.u. of a link of capacity images, as in the MTM. The only difference with the MTM is that different service‐classes may have different sets of thresholds. Specifically, we consider images service‐classes of Poisson arriving calls with mean arrival rates images that require images b.u. per call and a mean service time images, exponentially distributed. Calls compete for the available bandwidth under the CS policy. The offered traffic‐load of calls of service‐class images is images. Let images, and images. Each arriving call of a service‐class images may have images bandwidth and service‐time requirements, that is, one initial requirement with values images and images more requirements with values images, where images and images, and images. The pair images is used when images, where images and images are two successive thresholds of service‐class images, while images; the highest possible threshold (other than images) is images (see Figure 2.20). By convention, images and images, while the pair images is used when images.

Schematic displaying 2 vertical lines enclosed in a rectangle having rounded left and right sides. A word THRESHOLDS is found at the middle with dotted lines linking to the tick marks J12, J11, J10, J22, J21, and J20.

Figure 2.20 The CDTM principle of operation.

2.9.2 The Analytical Model

2.9.2.1 Steady State Probabilities

Similar to the MTM, the CDTM has no PFS. To describe the CDTM in steady state, the following LB equations are assumed:

(2.63)equation
(2.64)equation

where images denotes the mean number of service‐class images calls, with images, in state images.

Formulas (2.63) and (2.64) are graphically presented in Figure 2.22.

Similar to the analysis of the MTM and based on ( 2.63) and ( 2.64), we have the following recursive formula for the calculation of images [ 4]:

(2.65)equation

where images images  and images.

As a summary, in order to derive (2.65), the following assumptions (approximations) are necessary:

  1. (i) The assumption that the LB equations ( 2.63) and ( 2.64) do exist. This is the first source of error in ( 2.65).
  2. (ii) images is assumed negligible (i.e., zero) outside images. This assumption is the migration approximation, and we name the state space in which images, the migration space images. Let us recall that in the migration space, calls accepted in the system with other than the maximum bandwidth requirement are negligible. This assumption is the second source of error in ( 2.65) and is represented by the variable images.
  3. (iii) images is assumed negligible (i.e., zero), if images and images. This is the upward migration approximation, and we name the state space in which images, the upward migration space. Let us recall that in the upward migration space, calls accepted in the system with their maximum bandwidth are negligible. This assumption is the third source of error in ( 2.65) and is represented by the variable images.

Note that in both (ii) and (iii), the values of images and images may not be negligible in the corresponding migration and upward migration spaces, respectively (see Example 2.18). The determination of these values can improve the accuracy of the CDTM, compared to simulation, but is beyond the scope of this book. The reader may refer to [7,8].

Left: 2 Circles labeled j-bk and j linked by right and left arrows labeled λk and μkyk(j), respectively. Right: 2 Circles labeled j-bkcs and j linked by right and left arrows labeled λk and μkcsykcs( j), respectively.

Figure 2.22 Graphical representation of the LB equations 2.63 (left) and 2.64 (right).

An excerpt from a state transition diagram displaying circles labeled j = 1, j = 2, j = 3, j = 4, j = 5, and j = 6 enclosed in overlapping boxes, with j = 6 linking to j = 4 and j = 5, j = 5 linking to j = 2, and j = 2 linking to j = 6.

Figure 2.23 Migration and upward migration spaces (Example 2.18).

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

Based on ( 2.65), we can determine the following performance measures:

  • The CBP of service‐class images calls with images b.u., images, via:
    (2.66)equation
    where images is the normalization constant.
  • The CBP of service‐class images calls with images b.u., images, via (1.66).
  • The conditional CBP of service‐class images calls with images given that images, images, via:
    (2.67)equation
  • The link utilization, images, via (1.54).
  • The mean number of service‐class images calls with images b.u. in state images, via:
    (2.68)equation
    where images
  • The mean number of service‐class images calls with images b.u. in state images, images, via:
    (2.69)equation
  • The mean number of in‐service calls of service‐class images accepted in the system with images, via ( 2.16).
  • The mean number of in‐service calls of service‐class images accepted in the system with images, via ( 2.56).

2.10 The Connection Dependent Threshold Model under the BR Policy

2.10.1 The Service System

In the CDTM under the BR policy (CDTM/BR), images b.u. are reserved to benefit calls of all other service‐classes apart from service‐class images. The application of the BR policy in the CDTM is similar to that of the MTM/BR.

2.10.2 The Analytical Model

2.10.2.1 Link Occupancy Distribution

Similar to the MTM/BR, we adopt the Roberts method for the calculation of images in the CDTM/BR. Based on the Roberts method, ( 2.65) takes the form [ 4]:

(2.70)equation

where images images images and images.

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

Based on (2.70), we can determine the following performance measures:

  • The CBP of service‐class k calls with images b.u., images, as follows [ 4]:
    (2.71)equation
    where images is the normalization constant.
  • The CBP of service‐class images calls with images b.u., images, via (1.66).
  • The conditional CBP of service‐class images calls with images given that images, via:
    (2.72)equation
  • The link utilization, images, via (1.54).
  • The mean number of service‐class images calls with images b.u. in state images, via:
    (2.73)equation
    where images
  • The mean number of service‐class images calls with images b.u. in state images, via:
    (2.74)equation
    where images.
  • The mean number of in‐service calls of service‐class images accepted in the system with images, via ( 2.16).
  • The mean number of in‐service calls of service‐class images accepted in the system with images, via ( 2.56).

2.11 Applications

We concentrate only on the CDTM since it comprises the retry and threshold models. The initial motivation for the CDTM was the available bit rate (ABR) service of asynchronous transfer mode (ATM) networks. The ABR service is a purely elastic service in which the notion of equivalent bandwidth is not well applicable (i.e., the ABR service cannot be considered a stream service having its average bandwidth per call, as constant rate). Therefore, a different model to the EMLM is needed, and this is the CDTM because it sufficiently models an elastic call at its set‐up phase (but not during the entire call duration) by adequately setting the threshold parameters. Thus, the CDTM is applicable to any elastic service at the call set‐up phase, as long as it is not a bandwidth hungry application wasting all the available bandwidth. For this reason, a threshold scheme must be applied (e.g., Figure 2.20). The logic behind the threshold scheme is that even if the available bandwidth of a link is large enough, a CAC does not always waste it on one call only, but saves a part of it for sharing with the next calls.

The minimum and the maximum bandwidth requirements of an elastic call are important CDTM parameters for the CBP calculation no matter what the thresholds are. The minimum bandwidth requirement is critical for the CBP value. If the minimum required bandwidth is zero, an elastic call should wait for any available bandwidth to start servicing. If the network (CAC) ignores the details of bandwidth requirements (i.e., the threshold scheme) of an elastic call, the assigned bandwidth will not meet the real needs of the call and the CBP calculation will not be accurate. Suppose, for instance, that an elastic call has the following thresholds scheme in a transmission link with bandwidth capacity of 19.2 Mbps: maximum rate of 1.536 Mbps for available link bandwidth at least 6.4 Mbps (first threshold at images Mbps), rate of 768 kbps for available link bandwidth at least 3.2 Mbps at least (second threshold at images Mbps), and minimum rate of 384 kbps for available link bandwidth less than 3.2 Mbps (third threshold at 19.2 Mbps, or at images Mbps). Assume that the CAC knows only the minimum and maximum resource requirements of this elastic call, and offers to it (a) 700 kbps or (b) 1.536 Mbps when the available bandwidth is 4.0 Mbps at least in both cases. In the first case, the holding time will be estimated incorrectly (by taking into account 768 kbps instead of 700 kbps), while in the second case, although the holding time will be estimated correctly, the threshold scheme has been violated, therefore the CBP through the CDTM cannot be accurate. As far as the number of thresholds between the minimum and the maximum bandwidth requirements is concerned, several values exist since bandwidth is quantized and provided as a group of b.u. (trunks). So, in a realistic network environment the number of thresholds is manageable.

In what follows, we concentrate on the applicability of the CDTM to WCDMA networks. We have skipped straightforward applications, albeit some of them are very interesting, such as the application of the CDTM on smart grid, for a fine control of energy consumption [9].

Applicability of the CDTM to WCDMA Networks

The CDTM can be applied to WCDMA networks (in the uplink) in a similar way to the EMLM. A single BS controlling a cell can be modeled as a system of certain bandwidth capacity. The b.u. can be an equivalent bandwidth defined by the load factor introduced, for instance, by a lower rate service‐class (e.g., voice). The load factor is determined by the signal‐to‐noise ratio (SNR), data rate, and activity factor (probability that a call is active – transmits) of the associated service‐class. As far as the inter‐cell interference is concerned, it is assumed log‐normally distributed1 and independent of the cell load. A call is accepted for service as long as there are enough resources available in the cell. The CAC policy is based on the estimation of the increase in the total interference (intra‐ and inter‐cell interference plus thermal noise) caused by the acceptance of new calls. After call acceptance, the SNR of all in‐service calls deteriorates; because of this, WCDMA systems usually have no hard limits on call capacity. A call should not be accepted if it increases the noise of all in‐service calls above a tolerable level. Poisson arriving calls to a cell may have several contingency resource/QoS requirements.

Let us consider that the QoS offered to each service‐class images belongs to one out of images alternative QoS levels, which depend on the occupied cell resources. In what follows, a service‐class images call of QoS level images is referred to as service‐class images call. A service‐class images call is characterized by the following QoS parameters: (i) images, transmission bit rate, (ii) images, mean service time (exponentially distributed), and (iii) images, BER parameter.

The application of the CDTM for the call‐level performance evaluation of WCDMA networks is necessary when assuming that a WCDMA cell accommodates not only stream service‐classes but also elastic service‐classes, which are associated with individual sets of thresholds (indicating the occupied cell resources). A variation of an elastic service‐class is an adaptive service‐class, in which calls may reduce their resources/bandwidth, but their service time is kept fixed. We can therefore consider three types of service‐classes:

  • Stream type: service‐classes that have only one QoS level (images).
  • Elastic type: service‐classes that have more than one QoS level (images) and the call's mean service time strongly depends on the QoS level (it holds: images.
  • Adaptive type: service‐classes that have more than one QoS level (images) and the call's mean service time is the same for all QoS levels (it holds: images.

Upon their arrival, elastic or adaptive calls select one resource requirement according to an associated threshold scheme; a resource requirement is not altered during the service‐time. For example, a QoS level can be assigned to an elastic service‐class images call at the arrival time and is based on the occupied system resources (cell load images), which is indicated through thresholds. The thresholds of an elastic service‐class images are denoted by images. The QoS level assignment is performed as follows. If images, then the elastic call is assigned the first QoS level images and occupies images system resources for an exponentially distributed service time with mean images. The symbol images comes from the load factor (see 2.78). If images, then the call is assigned to the second QoS level and occupies images resources for an exponentially distributed service‐time with mean images, and so on. Finally, if images, then the call is assigned to the images QoS level and occupies images resources for an exponentially distributed service time with mean images. We assume that an elastic call has a certain amount of data to transmit. Therefore, a call's service time should be conversely proportional to the allocated resources. For this reason, the mean call service times, images, are chosen so that the product images remains constant for every QoS level images. As far as the offered traffic‐load of service‐class images calls is concerned, it is defined as images.

Interference and Call Admission Control

We assume perfect power control, i.e., at the BS, the same amount of power, images, is received from each service‐class images call. Since in WCDMA systems all users transmit within the same frequency band, a single user sees the signals generated by all other users as interference. Intra‐cell interference, images, is caused by users of the cell and inter‐cell interference, images, is caused by users of the neighbouring cells. An amount of power images is due to thermal noise in the cell and corresponds to the intra‐cell interference when the cell is empty.

The CAC is performed by measuring the noise rise, images, which is defined as the ratio of the total received power at the BS, images, to the thermal noise power, images:

(2.75)equation

When a new call arrives, the CAC estimates the noise rise and if it exceeds a maximum value, images, the new call is blocked and lost.

Load factor and cell load

The cell load, images, is defined as the ratio of the received power from all active users to the total received power:

(2.76)equation

From (2.75) and (2.76), the relation between the noise rise and the cell load is:

(2.77)equation

The maximum value of the cell load, images, is the cell load which corresponds to the maximum noise rise, images2: images.

The resource/bandwidth requirement of a service‐class images call is expressed by the load factor, images [10]:

(2.78)equation

where images is the chip rate of the WCDMA carrier.

The cell load images can be written as the sum of the intra‐cell load, images (cell load derived from the active users of the reference cell), and the inter‐cell load, images (cell load derived from the active users of the neighbouring cells), i.e., images. The values of images and images are given by 2.79 and 2.80, respectively:

(2.79)equation

where images is the number of active users of service‐class images, while

(2.80)equation

For a new service‐class images call acceptance, the following condition must hold in the BS:

(2.81)equation

That is, an arriving call with resource requirement images is accepted in the cell if and only if, after its acceptance, the cell load remains below images.

Local Blocking Probabilities

Due to (2.81), the probability that a new service‐class images call is blocked when arriving at an instant with intra‐cell load, images, is called the local blocking probability (LBP), images, and can be calculated by (based on ( 2.78)–( 2.80)):

(2.82)equation

In ( 2.80), the images can be modelled as a lognormal random variable (with parameters mean images and variance images), which is independent of the intra‐cell interference. Hence, the mean, images, and the variance, images, of images are calculated by:

(2.83)equation
(2.84)equation

Consequently, because of ( 2.80), the inter‐cell load, images, will also be a lognormal random variable. Its mean, images, and variance, images, are calculated as follows:

(2.85)equation
(2.86)equation

where the parameters images and images can be determined by solving (2.85) and (2.86):

(2.87)equation
(2.88)equation

where images is the coefficient of variation.

Thus, (2.82) can be rewritten as:

(2.89)equation

The RHS of (2.89), is the cumulative distribution function (CDF) of images. It is denoted by images and can be calculated from:

(2.90)equation

where images is the well‐known error function.

Hence, if we substitute images into (2.90), from ( 2.89) we can calculate the LBP of service‐class images calls as follows:

(2.91)equation

Parameters' Discretization

To apply the CDTM in WCDMA systems, parameters' discretization is required. It is achieved by the introduction of a basic cell load unit images (e.g., granularity of images, used in Example 2.23). The CDTM parameters of system capacity, the total number of occupied b.u. in the system, the assigned number of b.u. to an in‐service call, and a bandwidth threshold are obtained by discretizing the cell load, the maximum cell load, the load factor, and the resource threshold, respectively:

(2.92)equation

Incorporating the User Activity and LBP

The user activity is described by the activity factor, images, which represents the fraction of the active period of a service‐class images call/user over the entire service time (images). In the CDTM, we consider that calls are active during the entire service time and we do not distinguish active users from passive users. However, in WCDMA systems it is essential to consider such a distinction because passive users do not consume any system resources. Hence, a system state images does not represent the total number of occupied b.u. Instead, it represents the total number of b.u. that would be occupied if all (mobile) users were active. Let images denote the total number of occupied b.u. at an instant. In the CDTM, images is always equal to images, while in WCDMA networks we have images. When all users are passive, images, while images when all users are active.

The bandwidth occupancy, images, is defined as the conditional probability that images b.u. are occupied in state images and, for user activity images, it can be calculated recursively by:

(2.93)equation

where images is the highest reachable system state, images for images, and images is called resource/bandwidth share and denotes the proportion of the total occupied resources, images, from service‐class images calls (see (2.95)).

In WCDMA systems, due to the intra‐/inter‐cell interference, blocking of a service‐class images call may occur at any state images with a probability images. This probability, called the local blocking factor (LBF), is calculated by summing over images the LBP multiplied by the corresponding bandwidth occupancies:

(2.94)equation

where for images use (2.91) with images. Note that when images, images (since in this case images and images).

The service‐class images bandwidth share in state images (requiring images b.u.) is derived assuming LB between adjacent systems states, while incorporating the LBF and the parameter delta (indicating the upward migration and migration approximations):

(2.95)equation

where images, while for images:

(2.96)equation

State Probabilities and CBP

The state probabilities are given by:

(2.97)equation

for images and images  for images, with images.

The CBP of service‐class images can be calculated by adding all the state probabilities multiplied by the corresponding LBFs:

(2.98)equation

Due to the contingency bandwidth requirements, images, we need also to sum over images in specific areas defined by thresholds. This is done with the aid of the parameter gamma:

(2.99)equation

2.12 Further Reading

Extensions of the retry or thresholds models are categorized in wired [11–18], wireless [19–21], and optical networks [22]. In [ 11], the threshold models are extended to include BPP traffic. The CBP calculations are based either on recursive formulas or on convolution algorithms. In [12] and [13], the single threshold of the STM is replaced by two thresholds. When a new call finds the occupied link bandwidth above a threshold, it can be accepted in the link with its lower bandwidth requirement (similar to the STM). When the occupied link bandwidth becomes less than the second threshold then an in‐service call (accepted with its lower bandwidth requirement) can increase its bandwidth to its peak‐bandwidth requirement. In [14] and [15], the CDTM is extended to allow call bandwidth compression/expansion of in‐service calls with [ 15] or without [ 14] the existence of the BR policy (more on the subject of bandwidth compression/expansion and Poisson arriving calls can be found in Chapters and ). In [16– 18], a variant of the SRM/STM is proposed. Specifically, some service‐classes are characterized cooperative and the rest non‐cooperative. Users from a cooperative service‐class can retry with a certain probability to be connected in the system with reduced bandwidth when blocked with their initial peak‐bandwidth and the total occupied bandwidth of the system is below a threshold. This behavior increases the QoS perceived by other users. In [ 19], the threshold models are extended to include the CBP calculation in the uplink of a UMTS network. To this end, the notion of local (soft) blocking is incorporated in the model. The latter means that a call may be blocked in any state of the system if its acceptance violates the QoS, in terms of noise, of all in‐service calls (see also [23–26]). In [20], the threshold models are extended for the call‐level analysis of the Iub interface in UMTS networks. In [ 21], a multi‐threshold teletraffic model for heterogeneous CDMA networks is proposed. The model enables QoS differentiation of handover traffic when elastic and adaptive service‐classes are present. Furthermore, an applicability framework is proposed that takes into account advances in Cloud‐RAN and self‐organizing network (SON) technologies. In [ 22], the CDTM is extended for the calculation of connection failure probabilities (due to unavailability of a wavelength) and CBP (due to the restricted bandwidth capacity of a wavelength) in hybrid TDM‐WDM PONs with DWA.

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