In discrete-time Markov chains, the time that the system spends in the same state is geometrically distributed [2]. We can easily prove this statement. Let us assume that the system has entered a state i. Then, the probability that the system will remain in the same state is p ii . The probability that the system will leave its state at the next step is (1 – p ii ). Due to the memoryless property of the Markov chains, we may write the following: P{system remains in state i after m consecutive steps} = (1 – p ii )p ii m (4.27) For continuous-time Markov chain, we have exponential distribution of the time in single state (discrete-state continuous-time Markov process, see Figure 4.3), and we may write the following: () { }Ft P t e t =≤=− − ξ λ 1 (4.28) where λ is a parameter of the exponential distribution. The density function of the exponential distribution (Figure 4.4) is given by ()ft e t = − λ λ (4.29) The probability that the interarrival time between two consecutive arrivals will be up to t after it was t 0 may be calculated by Teletraffic Theory 99 0 0.1 0.2 0.3 0.4 0.5 02468101214161820 Time lambda=0.1 lambda=0.3 lambda=0.5 Probability density function Figure 4.3 Probability density functions of discrete-state continuous-time Markov chain. {} [] {} {}{} {} Pttt Pt t t Pt PttPt Pt ξξ ξ ξ ξξ ξ ≤+ > = <≤+ > = ≤+ − ≤ −≤ = 00 00 0 00 0 1 1 () () () −−− =− −+ − − − ee e e tt t t t λ λ λ λ 0 0 0 1 1 (4.30) If there is only one event from t = 0 to time t = t 0 , then the probability for a new event to occur in next time period t (from t 0 to t + t 0 ) does not depend upon t 0 . We will further apply Markov processes in telecommunications because most of the random events can be considered in a Markov chain fashion. 4.4 The Birth-Death Process The birth-death process is a special case of the Markov processes. Here, the tran - sitions are permitted between adjacent states only. We are mainly interested in continuous-time processes, so we consider birth-death processes in that fashion. The probability that more then one event will occur in an infinitesimal time interval is zero: {} lim P t t > = → 1 0 0 ∆ ∆ (4.31) This feature is called ordinarity. We usually write 100 Traffic Analysis and Design of Wireless IP Networks 0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0 2 4 6 8101214161820 Time f(t) f(t-8) Probability density function Figure 4.4 Probability density function of the exponential distribution. {} () P t ot > = 1 ∆ ∆ (4.32) We consider only continuous-time birth-death process, which are of pri - mary interest to us in this book. Birth-death processes are often used for the analysis of mass systems in telecommunications and computer networks (which means a large number of users in the system). They are often appropriate for modeling changes in the size of population. In a telecommunications network the population is the number of users in the system. Therefore, we can refer to some state E k according to the number of users. So, without losing generality, we may denote with E k the state of the system when the population is of size k. From the state k, birth-death process may transit only in state k + 1 and state k – 1, or remain in the state k during time interval ∆t. We introduce the notion of birth rate λ k as well as death rate µ k in a state k. Due to the memoryless property of the birth-death process, these birth and death rates are independent of time, but depend upon the current state E k only. Possible transitions in a birth-death process are shown in Figure 4.5. Because the birth-death process permits transitions among neighboring states only, we can describe it by a state transition diagram shown in Figure 4.6. We will refer to this type of diagram as a one-dimensional Markov chain. Teletraffic Theory 101 k 1+ k k– 1 k Time t Death Birth No change tt +∆ Figure 4.5 State transitions in a birth-death process. 01 2 λ 0 µ 1 i λ 1 λ 2 λ i –1 λ i µ 2 µ 3 µ i µ i +1 Figure 4.6 One-dimensional Markov chain for a birth-death process and infinite population in the system. The state E k can be reached in time interval ∆t from states E k–1 , E k , and E k+1 . Considering that birth and death are independent and using Figure 4.5, we may write: 1. The probability of exactly one birth in (t, t +∆t) when the process is in state E k-1 is λ k–1 ∆t + o(∆t). 2. The probability of exactly one death in (t, t +∆t) when the process is in state E k+1 is µ k+1 ∆t + o(∆t). 3. The probability of exactly zero births in (t, t +∆t) when the process is in state E k is1–λ k ∆t + o(∆t). 4. The probability of exactly zero deaths in (t, t +∆t) when the process is in state E k is1–µ k ∆t + o(∆t). Let us denote with p ij the probability for a transition from state i to state j. Using the Kolgomorov-Chapman approach, we analyze possible transitions of our particle. In this case, in a time interval (t, t +∆t) we can enter state E k only by three mutually exclusive possibilities: 1. P{no state change occurred in state k} = [1 – λ k ∆t + o(∆t)] [1 – µ k ∆t + o(∆t)]; 2. P{the system was in state k – 1 and we had one birth} = λ k–1 ∆t + o(∆t); 3. P{the system was in state k + 1 and we had one death} = µ k+1 ∆t + o(∆t). If we use P k (t) to denote the probability that the system was in state k at time t, and we use p k,j (∆t) to denote the probability for a transition from state k to state j during time t, then we may write ( ) () ( ) () ( ) () ( ) Pt t Ptp t P tp t Ptp t kkkkkkk kkk += + + −− ++ ∆∆ ∆ ∆ ,, , 11 11 (4.33) If p i,j (∆t) are expressed using birth and death rates, we obtain ()() () () () () ( ) Pt t Pt tPt tP t tP t o t kkkkk kk kk += −+ ++ + −− ++ ∆∆ ∆∆∆ λµ λµ 11 11 (4.34) From the last equation, with some algebra, we may write 102 Traffic Analysis and Design of Wireless IP Networks ()() () () () () ( ) Pt t Pt t Pt P t Pt ot kk kkk kk kk +− =− + + ++ −− ++ ∆ ∆ ∆ λµ λ µ 11 11 (4.35) If we allow ∆t→0, then we have () () () () () () dP t dt Pt Pt Ptk dP t dt k kkk kk kk =− + + + ≥ =− −− ++ λµ λ µ λ 11 11 0 0 1, () ()Pt Pt k 011 0+=µ , (4.36) We have lower boundary at the state 0 (no population). Also, it is possible to have an upper boundary if we have specified the maximum number of users in the system. Because birth-death process is a special case of the Markov processes, we can apply the same matrix notation introduced by (4.11) to (4.13), and we obtain the following transition matrix: () () T = − −+ −+ λλ µµλ λ µµλλ 00 111 1 2222 000 00 00                 (4.37) Let us now assume, for simplicity, that the system starts at state E 0 at time t = 0: ()P k k k 0 10 00 = = >    , , (4.38) From (4.36) and (4.38) we get a differential equation, the solution to which is ()Pt e t 0 = − λ (4.39) Then, it is easy to continue for values k≥1: () () Pt t k ekt k k t =≥≥ − λ λ ! ,,00 (4.40) Teletraffic Theory 103 The last relation is called Poisson distribution. It characterizes the Poisson process, which, in fact, is a pure birth process with λ λ µ kk k k k= ≥ <    = , , 0 00 0and for every (4.41) The Poisson process is significant in traffic theory in telecommunications, especially for circuit-switched networks, as we shall see later in this chapter. But the Poisson process has an even wider significance. It was shown by Palm that in many cases a large sum of independent stationary renewal processes tends to a Poisson process. 4.4.1 Stationary System In practice we are interested in a stationary regime of processes because it is con- venient for the analysis due to unique distribution of the state probabilities, independent of the initial condition. For a stationary system it holds that ()PPt k t k = →∞ lim (4.42) Now, we may write () lim t k dP t dt →∞ = 0 (4.43) We define a system that satisfies the last equation as a system in statistical equilibrium. Furthermore, by using (4.36) we obtain () λµ λµ kk kk k kk PP Pk −− ++ +−+=≥ 11 11 01, (4.44) λλ 11 00 00PP k−==, (4.45) Needless to say, because we cannot have a negative number of users in the system, p i = 0 for i < 0, and λ i = 0, i < 0; µ j = 0, j < 1. Because the birth- death process in statistical equilibrium is a case of Markov processes, we can apply the general equation for a Markov chain in equilibrium, which can be derived directly from the state-diagram given in Figure 4.7. To obtain dependences among state probabilities, we draw arbitrary boundaries as shown in Figure 4.7. The total outgoing rate from a closed boundary should be equal to the total incoming rate into the boundary in a state of equilibrium. Using boundary 1 from Figure 4.7, we obtain 104 Traffic Analysis and Design of Wireless IP Networks TEAMFLY Team-Fly ® µλ 11 00 PP= (4.46) and so on; from ith boundary we get λµ ii ii PP −− = 11 (4.47) Also, we require the conservation relation to hold for state probabilities: P i i = = ∞ ∑ 1 0 (4.48) From (4.46) and (4.47), after some simple algebra, we obtain PP k i i i k = + = − ∏ 0 1 0 1 λ µ (4.49) Then, if we apply (4.48) and (4.49), we obtain P i i i k k 0 1 0 1 1 1 1 = + + = − = ∞ ∏∑ λ µ (4.50) Because for all probabilities must hold 0 ≤ P k ≤ 1, there is a restriction on the values of the rates λ k and µ k+1 , k = 0, 1, . To address the existence of the state probabilities P k , we define two sums as follows: S i i i k k 1 1 0 1 1 = + = − = ∞ ∏∑ λ µ (4.51) S P P k i i i k kk 2 01 0 1 11 == + = − = ∞ = ∞ ∏∑∑ µ λ (4.52) Teletraffic Theory 105 0 1 2 λ 0 µ 1 i λ 1 λ 2 λ i –1 λ i µ 2 µ 3 µ i µ i +1 Boundary i Boundary 1 Figure 4.7 State-diagram of a birth-death process in equilibrium. All states will be ergodic only and only if S 1 < ∞ and S 2 = ∞ . Because we need an ergodic process to have equilibrium, it is of most interest to our analysis. The condition is fulfilled if there exists some k 0 such that for all k > k 0 it holds that λ µ κ+1 k <1 (4.53) This condition is usually true in telecommunications systems that we design. 4.4.2 Birth-Death Queuing Systems in Equilibrium Let us consider the importance of the statistical equilibrium of a birth-death process. We can define two types of equilibrium: a global balance and a local balance. Global balance may be defined by using (4.44) and by applying it in infini- tesimal time interval ∆t: λµλ µ kk kk k k k k tP tP tP tP k∆∆ ∆ ∆+= + ≥ −− ++11 11 1 (4.54) By analyzing the last equation, we may observe that the left side of the rela- tion gives the probability of a transition to neighboring states with respect to k—that is, to k + 1 (a new birth), and to k – 1 (a death). The right side of (4.54) gives the probabilities of transition from adjacent states to the state k.Wecan say that total outgoing traffic intensity from a particular state k is equal to the total incoming traffic intensity to that state. This is referred to as a global balance. Local balance is defined by multiplying (4.47) by ∆t, which leads to λµ ii ii tP tP i −− == 11 123∆∆for , , , (4.55) From the last equation it is obvious that the possibility of a transition from state k – 1 to state k (the left side) is equal to the transition probability in the reverse direction (the right side of the equation). This is called a local balance, and (4.55) is referred to as a local balance relation. 4.5 Teletraffic Theory for Loss Systems with Full Accessibility We covered the basics of queuing theory in previous sections of this chap - ter. Now, let us go through traditional way of design and analysis in telecommu - nications represented by the famous Erlang’s loss formula (elsewhere it is referred to as Erlang-B formula or Erlang’s first formula). 106 Traffic Analysis and Design of Wireless IP Networks In the early decades of telephony and switching systems, Erlang made an extensive analysis of the traffic data such as telephone calls initiated by users connected to a switching system (telephone exchange), blocking of the calls and their duration. He found that call arrivals suit well into a Poisson process. Also, call duration was shown to be easily modeled by using the exponential distribu - tion for the call duration times. According to the above statement, we may say that Erlang’s loss formula is based on the following model: • Arrival process is Poisson and service times are exponentially distributed. • We consider a circuit-switched system with servers (channels, trunks, or time slots) working in parallel. • An arrival is accepted for service if any channel is idle. The system allo - cates one channel per call. We say the group (of channels) has full acces - sibility when every incoming user competes with other users for all idle channels (not allocated resources). Using the queuing theory and Kendall notation for queuing systems [2], we can describe Erlang’s conclusions by using M/M/n/n queuing system. In this case n servers are n channels that may serve up to n users at the same time. Usu- ally, the number of potential users is many times higher than the number of available channels (resources) due to the economic aspects of telecommunica- tions networks design. This statement holds for both analog and digital circuit- switched networks, because in both cases we have one type of traffic only (voice telephony) and the system allocates equal resources for each call. Of course, because telephony is bidirectional, we have occupancy of two channels per call (one for each direction), but in traditional traffic theory it is enough to consider only one direction in calculations due to symmetrical resource allocation in both directions. This picture will change, however, if we introduce packet-based communication and heterogeneous services, as we shall later see. The state diagram for M/M/n/n is given in Figure 4.8. For this system we have λλ, µµ, k k kn kk n == − == 012 1 12 , , , , , , , (4.56) Teletraffic Theory 107 01 2 λ µ n– 1 λ λ λ λ 2 µ 3 µ ( –1) n µ n µ n Figure 4.8 State-diagram for Erlang’s loss formula. Using (4.49) and (4.50), which we proved for the general case of the birth-death processes, and by replacing λ k and µ k according (4.56), we obtain P k i A k A i k k i i n k i i n =             = = = ∑ ∑ λ µ λ µ ! ! ! ! 0 0 (4.57) where A = λ/µ is intensity of the offered traffic. It is expressed in units Erlangs in honor of Erlang. Relation (4.57) is called Erlang distribution or truncated Pois - son distribution. Definition of the carried traffic: We define the carried traffic per single channel as a sum of busy times for each channel during time interval T divided by the time interval. For a pool of resources, it is given by () A xtdt T T = ∫ 0 (4.58) where x is the average number of busy channels: () ()xt xP xt x n = = ∑ , 0 (4.59) If there are many users in the system, then we can apply the statistical equilibrium where the number of simultaneously occupied channels does not depend upon the moment in time—that is, P(x, t) = P(x). Then, the offered traffic may be calculated by using the following equation: ()YxPx x n = = ∑ 1 (4.60) Also, we may define offered traffic as average intensity of calls C A (calls/sec - ond) multiplied by average call duration time t µ : ACt A = µ (4.61) It is obvious that C A = λ, while t µ = 1/µ due to exponential distribution of the call duration, so we get 108 Traffic Analysis and Design of Wireless IP Networks [...]... 14 16 18 20 22 24 26 28 30 Offered traffic Figure 4. 9 Blocking as a function of offered traffic by using Erlang’s first formula 112 Traffic Analysis and Design of Wireless IP Networks integrated heterogeneous traffic sources are called integrated networks (it is a notation for a circuit-switched networks) Each service has different traffic characteristics In networks with asynchronous transport of. .. requested bandwidth units per call of service type i Then we may write 120 Traffic Analysis and Design of Wireless IP Networks c i j i ≤ n i ≤ n , i = 1, 2, , s s ∑ c i ji ≤ n (4. 99) i =1 where ji is number of calls from service type i 4. 6 .4 Priority Queuing So far, we have considered only classical queuing systems where all traffic processes are birth and death processes In systems with different traffic. .. to the amount of resources and their utilization Low utilization means bad economy Also, losing traffic Table 4. 1 Offered Traffic for a Given GoS ≤ 2% and Average Channel Utilization at Normal Traffic Load and at Overload of 30% N 1 10 50 100 A [Erlang] 0.02 5.08 40 . 24 87.95 0.0196 E1,N(A) ≤ 0.02 α= Y/N 0 .49 78 0.7887 0.8619 A’ = 1.3*A [Erlang] 0.026 6.60 52.31 1 14. 34 E1,N(A’) 0.0253 0.06 34 0.1306 0.1626... , λk, and discrete positive weights W(λ) (i.e., p1, p2, , pk) where k ∑p j =1 j =1 then we can write the hyper-exponential distribution function as (4. 86) 116 Traffic Analysis and Design of Wireless IP Networks k F (t ) = 1 − ∑ p j e − λ jt (4. 87) j =1 For example, hyper-exponential distribution can be used for modeling the call-holding time of a system with multiple traffic types, where each traffic. .. function of the equipment be ge(n), a function of the available resources (we consider channels without loosing the generality of the approach) The total cost of the system for a given number of channels n is a sum of the cost of resources and the costs due to lost traffic: 90 80 70 Total costs Costs 60 Optimum 50 40 30 Costs due to lost traffic 20 Costs of resources 10 0 0 10 20 30 40 50 Number of channels... access networks at either side and on the core network In 132 Traffic Analysis and Design of Wireless IP Networks circuit-switched networks we usually use more stringent GoS (e.g., GoS≤0.1%) for the interconnection backbone network (e.g., trunks between two telephone exchanges) than for the access networks The efficiency of the transmission is much higher on the backbone networks due to accumulation of. .. due to phenomenon of handover process Teletraffic theory for wireless networks given in this chapter is targeted to circuit-switched mobile networks such as 2G However, wireless LANs and 3G mobile networks experience heterogeneous and packet-based traffic Although modeling of realtime services (e.g., telephony) in these wireless networks can be performed using the traditional teletraffic theory, some... Web browsing and video streaming require different approaches (we refer to them in the following chapters) Finally, we introduced the fundamental principles in the design of telecommunication networks, which holds either for wireline or wireless networks: Teletraffic Theory 133 the design of telecommunications networks is based on the balance between the offered quality of service and costs of the system... 0.9575 130 Traffic Analysis and Design of Wireless IP Networks reduces revenue We want to optimize the system to maximize the revenue To obtain such a relation we need to know the additional traffic that system can carry when we add new resources For a group of channels or slots we define an improvement function at offered traffic A and n channels in the system as follows: F n ( A ) = Y n +1 − Y n (4. 128)... customers of the same or higher priority that are present in the queue The second portion s2 is the amount of higher priority work arriving during s1 The third portion is the amount of higher priority work arriving during s2, and so on Then, the mean waiting time is Work (time) s1 s2 s3 s4 Time Arrivals of type 1 customers Figure 4. 15 Waiting time of type 2 customer 122 Traffic Analysis and Design of Wireless . zero: {} lim P t t > = → 1 0 0 ∆ ∆ (4. 31) This feature is called ordinarity. We usually write 100 Traffic Analysis and Design of Wireless IP Networks 0 0. 04 0.08 0.12 0.16 0.2 0. 24 0.28 0 2 4 6 81012 141 61820 Time f(t) f(t-8) Probability. , = =− ∞ = − ∑∑ nP n knii n µ 0 1 (4. 75) 112 Traffic Analysis and Design of Wireless IP Networks Of course, the conservation of state probabilities must hold: P i i = = ∞ ∑ 1 0 (4. 76) By summing the equations (4. 75) from. distribution, which is a set of stochastic independent exponential distributions in parallel (Figure 4. 12). 1 14 Traffic Analysis and Design of Wireless IP Networks λ 1 λ k λ 2 Figure 4. 11 Steep distribution. TEAMFLY