10.1 The Erlang Multirate Loss Model with Batched Poisson Arrivals

10.1.1 The Service System

In the Erlang multirate loss model with batched Poisson arrivals (BP‐EMLM) we consider a link of capacity C b.u. that accommodates K different service‐classes under the CS policy. Calls of all service‐classes arrive in the link according to a batched Poisson process.

In the batched Poisson process, one basic principle of random arrivals, according to which no simultaneous arrivals occur (Figure 10.1a), is abolished, while another basic principle of random arrivals, according to which arrivals (the batches) occur at time‐points following a negative exponential distribution, is kept (Figure 10.1b) [1–4]. The batched Poisson process is important not only because in several applications calls arrive as batches (groups), but also because it can represent, in an approximate way, arrival processes that are more “peaked” and “bursty” (expressed by the peakedness factor ) than the Poisson process. The peakedness factor, , is the ratio of the variance over the mean of the number of arrivals: if , the arrival process is Poisson, if , the arrival process is quasi‐random (see Chapter 6), and if , the process is more peaked and bursty than Poisson. For example, batched Poisson processes are used to model overflow traffic (where ) [3].

Calls of service‐class require b.u. to be serviced and have an exponentially distributed service time with mean . Calls arrive to the link according to a batched Poisson process with arrival rate and batch size distribution , where denotes the probability that there are calls in an arriving batch of service‐class .

The batch blocking discipline adopted in the BP‐EMLM is partial batch blocking. According to the partial batch blocking discipline, a part of the entire batch (one or more calls) is accepted in the link while the rest of it is discarded when the available link bandwidth is not enough to accommodate the entire batch. More precisely, if an arriving batch of service‐class contains calls and the available link bandwidth is b.u., where , then calls will be accepted in the link (where is the largest integer not exceeding ) and the other calls will be blocked and lost. The complete batch blocking discipline, where the whole batch is blocked and lost, even if only one call of the batch cannot be accepted in the system, is not covered here since it results in complex and inefficient formulas for the calculation of the various performance measures [25–7].

Let be the number of in‐service calls of service‐class in the steady state and , . Due to the CS policy, the set of the state space is given by (I.36). In terms of and due to the partial batch blocking discipline (where each call is treated separately to the other calls of a batch), the CAC is identical to that of the EMLM (see Section 1.2.1).

10.1.2 The Analytical Model

10.1.2.1 Steady State Probabilities

In order to analyze the service system, the first target is to determine the PFS of the steady state probability . In the classical EMLM, the PFS of (1.30) is determined as a consequence of the LB between adjacent states. In the BP‐EMLM, the notion of LB as described in the EMLM does not hold; this is shown by the following example.

Based on (10.2) and (10.3), we have the following local flow balance equation across the level for service‐class calls:

(10.6)

The following lemma establishes the sufficiency of the flow balance across the levels for ensuring GB for state [ 3]:

The PFS that satisfies (10.6) and (10.10) is given by [ 3]:

(10.14)

where is the normalization constant, , is the state space of the system, , and

(10.15)

where is the offered traffic‐load (in erl).

An important batch size distribution is the geometric distribution, which is memoryless and a discrete equivalent of the exponential distribution. If calls of service‐class k arrive in batches of size , where is geometrically distributed with parameter , i.e., with , then we have in (10.15), . In that case, the average batch size for service‐class batches will be .

The existence of a PFS for the steady‐state probabilities leads to an accurate recursive formula for the calculation of the unnormalized link occupancy distribution, [ 3]:

(10.16)

where is the largest integer less than or equal to and is the complementary batch size distribution given by .

The proof of (10.16) is similar to the proof of (1.39) and therefore is omitted.

If calls of service‐class arrive in batches of size , where is given by the geometric distribution with parameter (for all ) then the BP‐EMLM coincides with the model proposed by Delbrouck in [8], as shown in [ 1] and [ 3]. More precisely, since , ( 10.16) takes the form:

(10.17)

If for and for , for all service‐classes, then we have a Poisson process and the EMLM results (i.e., the Kaufman–Roberts recursion of (1.39)).

10.1.2.2 TC Probabilities, CBP, Utilization, and Mean Number of In‐service Calls

The following performance measures can be determined based on ( 10.16):

• The TC probabilities of service‐class , via the formula:
  (10.18)
  where is the normalization constant.
• The CBP (or CC probabilities) of service‐class , via the formula:
  (10.19)
  where denotes the average size of service arriving batches and is given by . In the case of the geometric distribution, ).
• The mean number of in‐service calls of service‐class , via (1.56), where the values of can be determined via:
  (10.20)

  where and if .

  Note that (10.20) is an intermediate result for the proof of ( 10.16) [ 3]. Indeed, by multiplying both sides of ( 10.20) with and taking the sum over all service‐classes, we have ( 10.16).

• The link utilization, , via (1.54).

10.2 The Erlang Multirate Loss Model with Batched Poisson Arrivals under the BR Policy

10.2.2 The Analytical Model

10.2.2.1 Link Occupancy Distribution

In the BP‐EMLM/BR, the unnormalized values of the link occupancy distribution, , can be calculated in an approximate way according to either the Roberts method (see Section 1.3.2.2) or the Stasiak–Glabowski method (see Section 1.3.2.3). Herein we present the first method. Both methods have been investigated in [9], where the conclusions show that (i) both methods provide satisfactory results compared to simulation results, (ii) the Roberts method is preferable when only two service‐classes exist in the link or when the offered traffic‐load is low, and (iii) the Stasiak–Glabowski method can be considered when equalization of blocking probabilities is required, whereas it is preferable when more than two services exist and the offered traffic‐load is high.

Based on Roberts method, ( 10.16) takes the form [ 9]:

(10.21)

where is determined via (1.65).

10.2.2.2 TC Probabilities, CBP, Utilization, and Mean Number of In‐service Calls

The following performance measures can be determined based on (10.21):

• The TC probabilities of service‐class , via the formula:
  (10.22)
  where is the normalization constant.
• The CBP (or CC probabilities) of service‐class , via ( 10.19).
• The mean number of in‐service calls of service‐class , via (1.56), where the values of can be determined via:
  (10.23)
• The link utilization, , via (1.54).

10.3 The Erlang Multirate Loss Model with Batched Poisson Arrivals under the Threshold Policy

10.3.2 The Analytical Model

10.3.2.1 Steady State Probabilities

In order to analyze the service system, the first target is to determine the PFS of the steady state probability . Following the analysis of Section 10.1.2.1, it is easy to verify that the GB equation of ( 10.10) does hold for a state , where . It can also be proved that the PFS of the BP‐EMLM/TH is given by (10.14) and ( 10.15), where ( 10.15) holds for [10].

To rely on ( 10.14) and ( 10.15) for determining the TC and CC probabilities or link utilization, state space enumeration and processing are required. To circumvent this problem, which is quite complex for many service‐classes and large capacity links, we present the following three‐step convolution algorithm by modifying the convolution algorithm of [11]:

10.3.2.2 TC Probabilities, CBP, Utilization, and Mean Number of In‐service Calls

The following performance measures can be determined based on (10.25)–(10.28):

• The TC probabilities of service‐class , via the formula:
  (10.29)

  The first sum in (10.29) refers to states where there is not enough bandwidth for service‐class calls. The other term refers to states , where available link bandwidth exists, but nevertheless blocking occurs due to the TH policy. The normalization constant in the second term of ( 10.29) refers to the value used in (10.28).

• The CBP (or CC probabilities) of service‐class , via the formula ( 10.19).
• The mean number of in‐service calls of service‐class , via (1.56), where the values of can be determined via:
  (10.30)

  where and if .

  The nominator of (10.30) shows the “transfer” of the population of service‐class calls from the previous states to state due to an arrival of a service‐class batch. The normalization constant refers to the value used in (10.28).

• The link utilization, , via (1.54).

10.4 Applications

An interesting application of the BP‐EMLM has been proposed in [12] for the calculation of various performance measures, including call blocking and handover failure probabilities in a LEO mobile satellite system (MSS). In [ 12], a LEO‐MSS of contiguous “satellite‐fixed” cells is considered, each having a fixed capacity of channels. Moreover, each cell is modeled as a rectangle of length (e.g., km in the case of the iridium LEO‐MSS [13]) that forms a strip of contiguous coverage on the region of the Earth. Next, a few common assumptions are made. LEO satellite orbits are polar and circular. MUs are uniformly distributed on the Earth surface, while they are considered as fixed. This assumption is valid as long as the rotation of the Earth and the speed of a MU are negligible compared to the subsatellite point speed on the Earth [14]. Moreover, either beam handovers, within a particular footprint, or handovers between adjacent satellites of the same orbit plane may occur. The system of these cells accommodates MUs that generate calls of service‐classes with different QoS requirements. Each service‐class call requires a fixed number of channels for its whole duration in the system. New service‐class calls arrive in the system according to a batched Poisson process with arrival rate and batch size distribution . Due to the uniform MUs distribution, new calls may arrive in any cell with equal probability. The cell that a new call originates is the source cell. Due to the call arrival process of new service‐class calls, we model the arrival process of handover calls of service‐class via a batched Poisson process with arrival rate and batch size distribution . The arrival of handover calls in a cell is as follows. Handover calls cross the source cell's boundaries to the adjacent right cell having a constant velocity of , where (approx. 26600 km/h in the iridium constellation) is the subsatellite point speed. An in‐service call that departs from the last cell (cell ) will request a handover in cell 1, thus having a continuous cellular network (Figure 10.9).

Based on the above, let be the dwell time of a call in a cell (i.e., the time that a call remains in the cell). Then, is (i) uniformly distributed in for new calls in their source cell and (ii) deterministically equal to for handover calls that traverse any adjacent cell from border to border. Based on (ii), expresses the interarrival time for all handovers subsequent to the first one. As far as the duration of a service‐class call (new or handover) in the system and the channel holding time in a cell are concerned, they are exponentially distributed with mean and , respectively.

To determine formulas for the handover arrival rate and the channel holding time with mean of service‐class calls, we define:

1. The parameter as the ratio between the mean duration of a service‐class call in the system and the dwell time of a call in a cell [ 14]:
   (10.31)
2. The time as the interval from the arrival of a new service‐class call in the source cell to the instant of the first handover. is uniformly distributed in with pdf [15]:
   (10.32)
3. The probabilities and express the handover probability for a service‐class call in the source and in a transit cell, respectively. These probabilities are different due to the different distances covered by a MU in the source cell and in the transit cells. More precisely, is defined as:
   (10.33)

   where is the service‐class call duration time (exponentially distributed with mean ).

   The residual service time of a service‐class call after a successful handover request has the same pdf as (due to the memoryless property of the exponential distribution). It follows then that can be expressed by:

   (10.34)

The handover arrival rate can be related to by assuming that in each cell there exists a flow equilibrium between MUs entering and leaving the cell. In that case, we may write the following flow equilibrium equation (MUs entering the cell = MUs leaving the cell):

(10.35)

where refers to the CBP of new service‐class calls in the source cell and refers to the handover failure probability of service‐class calls in transit cells. The LHS of (10.35) refers to new and handover service‐class calls that are accepted in the cell with probability and , respectively. The RHS of ( 10.35) refers to (i) service‐class calls that are handed over to the transit cell (depicted by ), (ii) new calls that complete their service in the source cell without requesting a handover (depicted by ), and (iii) handover calls that do not handover to the transit cell (depicted by ).

Equation ( 10.35) can be rewritten as:

(10.36)

To derive a formula for the channel holding time of service‐class calls, note that channels are occupied in a cell by either new or handover calls. Furthermore, channels are occupied either until the end of service of a call or until a call is handed over to a transit cell. Since the channel holding time is expressed as in the case of the source cell and in the case of a transit cell, then the mean value of for is given by:

(10.37)

Now let and be the probabilities that a channel is occupied by a new and a handover service‐class call, respectively. Then:

(10.38)

Based on (10.37) and (10.38), the channel holding time of service‐class calls (either new or handover) is approximated by an exponential distribution whose mean is the weighted sum of ( 10.37) (for ) multiplied by the corresponding probabilities (for ) and (for ):

(10.39)

Having determined the various input traffic parameters, we can analyze the LEO‐MSS via the BP‐EMLM assuming that each cell is modeled as a multirate loss system, where all calls compete for the available channels under the CS policy. To facilitate the description of the analytical model in [ 12], we distinguish new from handover calls and assume that each cell accommodates calls of service‐classes. A service‐class is new if and handover if .

10.5 Further Reading

Due to the close relationship between the models of this chapter and the EMLM, the EMLM/BR, and the EMLM/TH, the interested reader may refer to the corresponding section of Chapter 1 . In addition, due to the fact that the BP‐EMLM can be used for the analysis of overflow traffic, it may be a candidate analytical model for various multirate loss systems that carry overflow traffic (e.g., [16–19]). Other extensions of the BP‐EMLM that show the applicability of the model in wireless networks appear in [20–22]. In [ 20], the BP‐EMLM is used for the CBP determination in the X2 link of LTE networks. In [21], the BP‐EMLM is extended to include multiple access interference, both the notion of local (soft) and hard blocking, the user's activity, as well as interference cancellation for the calculation of congestion probabilities in CDMA networks. An extension of [ 21] that incorporates the BP‐EMLM/BR is proposed in [ 22].

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