The Cellular Concept System Design Fundamentals

Transcription

1 Wireless Information Transmission System Lab. The Cellular Concept System Design Fundamentals Institute of Communications Engineering National Sun Yat-sen University

2 Table of Contents Frequency Reuse Channel Assignment Strategies Handoff Strategies Prioritizing Handoffs Practical Handoff Considerations Interference and System Capacity Power Control Trunking and Grade of Service Improving Coverage and Capacity in Cellular Systems Trunking Theory 2

3 Wireless Information Transmission System Lab. Frequency Reuse Institute of Communications Engineering National Sun Yat-sen University

4 Cellular System Design Considerations To solve problems of spectral congestion and user capacity. Replacing a single, high power transmitter with many low power transmitters. Neighboring base stations are assigned different groups of channels so that the interference between base stations is minimized. Available Channels are distributed throughput the geographic region and may be reused as many times as necessary. With fixed number of channels to support an arbitrarily large number of subscribers. 4

5 Concepts of Frequency Reuse 5

6 Cellular Networks and Frequency Reuse One important characteristic of cellular networks is the reuse of frequencies in different cells. By reuse frequencies, a high capacity can be achieved. However, the reuse distance has to be high enough, so that the interference caused by subscribers using the same frequency (or an adjacent frequency) in another cells is sufficiently low. To guarantee an appropriate speech quality, the carrier-to-interference-power-ratio (CIR) has to exceed a certain threshold CIR min which is 9 db for the GSM system. 6

7 Hexagons Hexagonal cell shape has been universally adopted. The actual radio coverage of a cell is known as the footprint and is determined from field measurements or propagation prediction models. Base stations can be placed at: The cell center center-excited cells omni-directional antennas. The cell vertices edge-excited cells sectored directional antennas. 7

8 Reuse Factor Due to the fact that the hexagonal geometry has exactly six equidistant neighbors and that the lines joining the centers of any cell and each of its neighbors are separated by multiples of 60 degrees, there are only certain cluster sizes and cell layouts which are possible. Reuse Factor = i 2 +ij+j 2 ; i,j are non-negative integers. 8

9 Co-channel Neighbor Cells Move i cells along any chain of hexagons. Turn 60 o counterclockwise and move j cells. For example: N=19: i =3, j =2; N=12: i =2, j =2; N=7; i =2, j =1; 9

10 Wireless Information Transmission System Lab. Channel Assignment Strategies Institute of Communications Engineering National Sun Yat-sen University

11 Channel Assignment Strategies Fixed Channel Allocation Dynamic Channel Allocation Hybrid Channel Allocation Borrowed Channel Allocation 11

12 Fixed Channel Assignment Each cell is allocated a predetermined set of voice channels. Any call attempt within the cell can only be served by the unused channels in that particular cell. Probability of blocking is high. 12

13 Dynamic Channel Assignment Strategy Channels are not allocated to different cells permanently. Each time a call request is made, the serving base station requests a channel from the MSC. The MSC allocates a channel to the requested cell following an algorithm that takes into account the likelihood of future blocking within the cell, the frequency of use of the candidate channel, the reuse distance of the channel, and other cost functions. MSC only allocates a given frequency if that frequency is not presently in use in the cell or any other cell which falls within the minimum restricted distance of frequency reuse to avoid co-channel interference. 13

14 Dynamic Channel Assignment Strategy MSC has to collect real-time data on channel occupancy, traffic distribution, and radio signal strength indications (RSSI) of all channels on a continuous basis. Reduce the likelihood of blocking at the expense of increasing the storage and computational load. 14

15 Borrowing Strategy Modified from fixed channel assignment strategies. A cell is allowed to borrow channels from a neighboring cell if all of its own channels are already occupied. The MSC supervises such borrowing procedures and ensures that the borrowing of a channel does not disrupt or interfere with any of the calls in progress in the donor cell. 15

16 Wireless Information Transmission System Lab. Handoff Strategies Institute of Communications Engineering National Sun Yat-sen University

17 Handoff / Handover In a cellular network, the process to transfer the ownership of a MS from a BS to another BS. Handoff not only involves identifying a new BS, but also requires that the notice and control signals be allocated to channels associated with the new base station. Usually, priority of handoff requests is higher than call initiation requests when allocating unused channels. Handoffs must be performed successfully and as infrequently as possible and be imperceptible to the uses. 17

18 Handoff / Handover Handover Occasions Bad signal quality on current channel noise or interference Traffic overload in current cell load balancing Handover Indicator: The parameters to monitor to determine HO occasion RSSI, in ensemble average sense. Bit Error Rate (BER)/Packet Error Rate (PER), more accurate. 18

19 Handoff / Handover Need to specify an optimum signal level to initiate a handoff. Minimum useable signal for acceptable voice quality at the base station receiver is normally taken as between -90 dbm to -100 dbm. Δ = Pr handoff Pr minimum useable If Δis too large, unnecessary handoffs may occur. If Δis too small, there may be insufficient time to complete a handoff. 19

20 Illustration of a handoff scenario at cell boundary 20

21 Illustration of a Handoff Scenario at Cell Boundary Figure (a) demonstrates the case where a handoff is not made and the signal drops below the minimum acceptable level to keep the channel active. The dropped call event in figure (a) can happen when there is an excessive delay by the MSC in assigning a handoff or when the threshold Δ is set too small for the handoff time the system. Excessive delays may occur during high traffic conditions due to computational loading at the MSC or due to the fact that no channels are available on any of the nearby base stations. 21

22 Handoff / Handover During handoff, it is important to ensure that the drop in the measured signal level is not due to momentary fading and that the mobile is actually moving away from the serving base station. The base station monitors the signal level for a certain period of time before a hand-off is initiated. The time over which a call may be maintained within a cell, without hand-off, is called the dwell time. 22

23 Handoff in 1st Generation Cellular Systems Signal strength measurements are made by the base stations and supervised by the MSC. Base station monitor the relative location of each user. Locator receiver is used to determine signal strengths of users in neighboring cells and is controlled by the MSC. Based on the information from locator receiver, MSC decides if a handoff is necessary or not. 23

24 Handoff in 2nd Generation TDMA Systems Handoff decisions are mobile assisted. In mobile assisted handoff (MAHO), every mobile measures the received power from surrounding base stations and reports the results to the serving base station. MAHO enables the call to be handed over at a much faster rate. 24

25 Handoff or Handover Mobile Assistant Handover more efficient. GSM: MS monitors all BSs MS reports the measurements to the BS MSC makes decision USDC (IS-54/136): BSs monitor all MSs. When a MS is leaving the cell, the BS sends it a measurement order The MS begins its measurement and reports MSC makes the Handover decision. 25

26 Handover Algorithms (IS-95 vs. WCDMA) Basic IS-95 handover algorithm uses absolute threshold algorithm. WCDMA handover algorithm users relative threshold algorithm. 26

27 Absolute Threshold Handover Eb/No Th_Add Th_Drop (1) (2) (3) (4) (5) (6) Time Neighbor Set Candidate Set Active Set Neighbor Set 27

28 Basic IS-95 HO Algorithm 1. Pilot strength exceed T_Add. MS sends a Pilot Strength Measurement Message and transfers pilot to the Candidate Set. 2. BS sends a Handover Direction Message. 3. Mobile station transfers pilot to the Active Set and sends a Hanover Completion Message. 4. Pilot strength drops below T_Drop. MS starts the handover drop timer. 5. Handover drop timer expires. MS sends a Pilot Strength Measurement Message. 6. BS sends a Handover Direction Message. MS moves pilot from the Active Set to the Neighbor Set and sends a Handover Completion Message. 28

29 Problems with Absolute Threshold Algorithm Some locations in the cell receive only weak pilots (requiring a lower handover threshold). Some locations in the cell receive a few strong an dominant pilots (requiring a higher handover threshold). 29

30 Relative Threshold HO Ec/Io MS_Ec/Io Strongest Pilot in Active Set AS_Th AS_Th_Hyst AS_Th_Hyst Window_Add Window_Drop T_Add T_Drop Time MS AS MS 30

31 Active vs. Monitored Set Active Set (AS): User information is sent from all these cells and they are simultaneously demodulated and coherently combined. Monitored Set (MS): Cells, which are not included in the active set, but are monitored according to a neighboring list assigned by the UTRAN. 31

32 Soft Handover Algorithm (for Active Set limit = 2) Ec/No CPICH 1 ΔT ΔT ΔT AS_Th + AS_Th_Hyst AS_Th -AS_Th_Hyst AS_Rep_Hyst CPICH 2 CPICH 3 Cell 1 Connected 32 Event A Add Cell 2 Event B Replace Cell 1 with Cell 3 Time Event C Remove Cell 3

33 Intersystem Handoff Intersystem handoff happens when a mobile moves from one cellular system to a different cellular system. The MSCs involved in the two cellular systems are different. Compatibility between the two MSCs must be determined. 33

34 Prioritizing Handover Guard Channel Concept : Use reserved guard channel for handover. Disadvantage: Reducing the total carrier traffic. Queuing of Handover Requests: To prevent forced termination by queuing the request. Queuing of handoffs is possible due to the fact that there is a finite time interval between the time the received signal level drops below the handoff threshold and the time the call is terminated due to insufficient signal level. 34

35 Practical Handover Consideration Problem 1: Simultaneous traffic of high speed and low speed mobiles. Small cell high speed mobile frequent handoff large cell Reduce capacity Solution: Umbrella Cell - Cell Split or Hierarchical Cell Structure By using different antenna heights and different power levels, it is possible to provide large and small cells which are colocated at a single location. Small cell for low speed mobile Large cell for high speed mobile Need Strength Detection and Handoff control. 35

36 The Umbrella Cell Approach 36

37 Practical Handover Consideration Problem 2: Cell Dragging Caused by pedestrian users that provide a very strong signal to the base station. Often occurs in an urban environment when there is a line-ofsight (LOS) radio path between the subscriber and the base station. As the user travels away from the base station at a very low speed, the average signal strength does not decay rapidly and the received signal at the BS may be above the handoff threshold, thus a handoff may not be made. Creates a potential interference and traffic management problem. Solution: Careful arrangement of handoff threshold and radio coverage parameters. 37

38 Handoff Control Parameters Δ = Pr Pr handoff minimum useable Δ 6~12 db (AMPS) Δ 0~6 db (GSM) Typical time to make a handoff once the signal level is blow the handoff threshold : 10 sec (AMPS) 1~2 sec (GSM) The faster handoff process supports a much greater range of options for handling high speed and low speed users and provides the MSC with substantial time to rescue a call that is in need of handoff. 38

39 Handoff Miscellaneous Intra-frequency Handoff: handoffs in the same system and carrier. Inter-frequency Handoff: handoffs between same systems and different carriers. May be used for handoff between different cell layers of the multi-layered cellular network, when the cell layers use different carrier frequencies. Inter-system Handoffs: handoffs between different systems. Inter-frequency and inter-system handoffs may be used for coverage or load balancing reasons. 39

40 Frequency Utilization with WCDMA Operator band 15 MHz Power Another UMTS operator MHz MHz Another UMTS operator MHz MHz 3 cell layers Uplink: Downlink: MHz MHz Frequency 40

41 Intra-Frequency Handoff Hard Handoff: assign different radio channels during a handoff. Soft Handoff: the ability to select between the instantaneous received signals from a variety of base stations. Soft handoff exploits macroscopic space diversity provided by the different physical locations of the base stations. Softer Handover: A mobile station is in the overlapping cell coverage area of two adjacent sectors of a base station. 41

42 Wireless Information Transmission System Lab. Interference and System Capacity Institute of Communications Engineering National Sun Yat-sen University

43 Interference The major source limiting cellular system capacity comes from interferences (as oppose to noise). Interference has been recognized as a major bottleneck in increasing capacity and is often responsible for dropped calls. Major Types of Interference: Co-Channel Interference Adjacent Channel Interference Intra-Cell Type Inter-Cell Type 43

44 Co-channel Cells and Interference In a given coverage area, there are several cells that use the same set of frequencies. These cells are called co-channel cells. The interference between signals from co-channel cells is called co-channel interference. Unlike thermal noise which can be overcome by increasing the signal-to-noise ratio (SNR), co-channel interference can't be overcome by simply increasing the carrier power because an increase in carrier power increases the interference to neighboring co-channel cells. To reduce co-channel interference, co-channel cells must be physically separated by a minimum distance. 44

45 Hexagon r = 2 3 R 3 r = R 2 R r R ( r R) 3 3 Cell Area = 6 = R = 2 3r

46 Hexagon A cell centered at (u,v) ( u, v) = (2ri,2rj) i=3 x y = u cos(30 = u sin(30 o o ) = ) + v 3 u 2 u = + 2 v 46

47 Distance between (u a, v a ) and (u b, v b ) D = ( x x ) + ( y y ) ab a b a b Hexagon Distance 2 2 ( u u ) ( v v ) ( u u ) ( v v ) For special case (distance from origin) ua ub = ua ub + + va vb = + + a b a b a b a b = (2 r) ( ia ib) ( ja jb) ( ia ib)( ja jb) 2 2 D= 2r i + j + ij 47

48 Hexagonal Cluster Hexagonal Cluster Each cluster is surrounded by six similar clusters with the same orientation Each cluster has a total area equivalent to what can be called a super-hexagon" view a cluster as a hexagon" 2 2 D = 2r i + j + ij 48

49 Super-Hexagon Concept D=Frequency Reuse Distance D R R ' = N = = D 2 A Super Hexagon A Cell ( R ' ) ( 2 2 3r ) 2 2 i j i j = + + = Cluster Size 49

50 Co-Channel Reuse Ratio The signal to co-channel interference ratio is independent of the transmitted power and becomes a function of the radius of the cell (R) and the distance between centers of the nearest cochannel cells (D). Co-channel Reuse Ratio Q : The spatial separation between cochannel cells relative to the coverage distance of a cell. D = 2 r = 2 r N i 2 + j 2 + i j r 3 = R 2 = R 3N D Q = = 3N R 50

51 CO-Channel Reuse Ratio Small Q Small N Large Capacity Large Q Large N Small Level of Co-Channel Interference 51

52 Co-Channel Signal to Interference Ratio (SIR) SIR: Signal to Interference Ratio. Consider only first tier interference. Assuming all interfering BS's are equal-distance. S S = P i r = 0 I R 3N Worst Case SIR= (Mobile) 52 i= 1 i 0 i= 1 I i PR t R ( 3 N) = n n id 0 i0 Pd t n n n i Example: i 0 =6, n=4 N=3 SIR=11.3 db N=7 SIR=18.7 db P 0 d d 0 n

53 7-Cell Co-channel Signal to Interference Ratio An approximation of the exact geometry. 53

54 7-Cell Co-Channel Signal to Interference Ratio An Approximation n S R = I 2( D R) + 2( D+ R) + 2D n n n S 1 = I 2 Q Q Q ( ) ( ) 54

55 Adjacent Channel Interference Adjacent channel interference results from imperfect receiver filters which allow nearby frequencies to leak into the passband. Adjacent channel interference can be minimized through careful filtering and channel assignments. Channels are allocated such that the frequency separation between channels in a given cell is maximized. 55

56 Near-Far Effect A nearby transmitter (which may or may not be of the same type as that used by the cellular system) captures the receiver of the subscriber. Alternatively, the near-far effect occurs when a mobile close to a base station transmits on a channel close to one being used by a weak mobile. The base station may have difficulty in discriminating the desired mobile user from the close adjacent channel mobile. 56

57 Wireless Information Transmission System Lab. Power Control Institute of Communications Engineering National Sun Yat-sen University

58 Power Control for 2G Cellular Systems Power levels transmitted by subscriber unit are under control by the serving base stations. Power control is to ensure that each mobile transmits the smallest power necessary to maintain a good quality link on the reverse channel. Power control not only helps prolong battery life for the subscriber unit, but also dramatically reduces the reverse channel S/I in the system. 58

59 Power Control in 3G (WCDMA) Tight and fast power control is perhaps the most important aspect in WCDMA in particular on the uplink. Without it, a single overpowered mobile could block a whole cell. Near-Far problem of CDMA: A MS close to the base station may be overpowered and block a large part of the cell. Power control in WCDMA: Open-loop power control Close-loop power control Inner-loop power control Outer-loop power control 59

60 Open Loop Power Control in WCDMA Attempt to make a rough estimation of path loss by measuring downlink beacon signal. Disadvantage: Far too inaccurate, because fast fading is essentially uncorrelated between uplink and downlink, due to the large frequency separation of uplink and downlink band of the WCDMA FDD mode. Open-loop power control is used in WCDMA to provide a coarse initial power setting of the MS at the beginning of a connection. 60

61 Outer Loop Power Control in WCDMA To adjusts the target SIR setpoint in the BS according to the individual radio link quality requirements, usually defined as BER or FER. The required SIR for FER depends on the mobile speed, multipath profile, and data rate. Should the transmission quality is decreasing, the Radio Network Controller (RNC) will command the Node B to increase the target SIR. Outer loop power control is implemented in RNC because there might be soft handover combining. 61

62 Inner-loop Power Control in WCDMA Uplink Base station performs frequent estimates of the received Signal-to-Interference Ratio (SIR) and compares it to a target SIR. If the measured SIR is higher than the target SIR, the base station will command the MS to lower the power. If SIR is too low, it will command the MS to increase its power. The power control is operated at a rate of 1500 times per second. 62

63 Inner-loop Power Control in WCDMA Downlink Adopt same techniques as those used in uplink. Operate at a rate of 1500 times per second. There is no near-far problem in downlink. Purposes for downlink closed-loop power are: Provide a marginal amount of additional power to MS at the cell edge as they suffer from increased other-cell interference. Enhancing weak signals caused by Rayleigh fading at low speeds when other error-correcting methods (interleaving and error correcting codes) doesn't work effectively. 63

64 Wireless Information Transmission System Lab. Trunking and Grade of Service Institute of Communications Engineering National Sun Yat-sen University

65 Trunking System Trunking system: A mechanism to allow many user to share fewer number of channels. Not every user calls at the same time. Penalty: Blocking Effect. If traffic is too heavy, call is blocked!! Small blocking probability is desired. There is a trade-off between the number of available circuits and blocking probability. 65

66 Grade of Service (GOS) Erlang: The amount of traffic intensity carried by a channel that is completely occupied. For example, a radio channel that is occupied for 30 minutes during an hour carries 0.5 Erlangs of traffic. Grade of Service (GOS) is a measure of the ability of a user to access a trunked system during the busiest hour. GOS is typically given as the likelihood that a call is blocked, or the likelihood of a call experiencing a delay greater than a certain queuing time. 66

67 Definitions of Terms Used in Trunking Theory Set-up Time: The time required to allocate a radio channel to a user. Blocked Call: Call which can't be completed at time of request, due to congestion. Holding Time: Average duration of a typical call. Denoted by H (in second). Traffic Intensity: Measure of channel time utilization, which is the average channel occupancy measured in Erlangs. This is a dimensionless quantity and may be used to measure the time utilization of single or multiple channels. Denoted by A. 67

68 Definitions of Terms Used in Trunking Theory Load: Traffic intensity across the entire trunked radio system, measured in Erlangs. Grade of Service (GOS): A measure of congestion which is specified as the probability of a call being blocked (Erlang B), or the probability of a call being delayed beyond a certain amount of time (Erlang C). Request Rate: The average number of call requests per unit time. Denoted by μ second

69 Computation of GOS A u =λh A u : each user generate a traffic intensity of A u Erlang λ : average number of call request per unit time for each user. H : average duration of a call. A=UA u A: Total offered traffic intensity. U: Total users in a system. A c =UA u /C C : # of channels in a truncking system A c : traffic intensity per channel 69

70 Computation of GOS There are two types of trunked systems: No queueing for call requests: for every user who requests service, it is assumed there is no setup time and the user is given immediate access to a channel if one is available. If no channels are available, the requesting user is blocked without access and is free to try again later. Called blocked calls cleared. GOS: Erlang B formula determines the probability that a call is blocked. A queue is provided to hold calls which are blocked. If a channel is not available immediately, the call request may be delayed until a channel becomes available. Called blocked calls delayed. GOS: Erlang C formula gives the likelihood that a call is initially denied access to the system. 70

71 Trunked Systems with no Queuing for Call Requests blocked calls cleared Blocking Probability -- Erlang B Formula: Pr[ blocking] = C C A C! A k k k=0! = GOS AMPS cellular system is designed for a GOS of 2% blocking. There are infinite number of users. Call requests are memoryless; both new and blocked users may request a channel at any time. Service time of a user is exponentially distributed Traffic requests are described by Poisson model. Inter-arrival times of call requests are independent and exponentially distributed. 71

72 Trunked Systems with Queuing for Call Requests Blocked Calls Delayed Erlang C Formula the likelihood of a call is initially denied to access a channel: Pr[ delay C > 0] = C 1 k C A A A + C! 1 C k = 0 k! A Pr[ delay > t] = Pr[ delay > 0]Pr[ delay > t delay > 0] = Pr[ delay > 0]exp( ( C A) t H ) 72

73 Trunked Systems with Queuing for Call Requests Blocked Calls Delayed The average delay D for all calls in a queued system is given by: D H = Pr[ delay > 0] C The average delay for those calls which are queued is given by: H C A A 73

74 Erlang B Chart 74

75 Erlang C Chart 75

76 Trunking Efficiency Trunking Efficiency is a measure of the number of users which can be offered a particular GOS with a particular configuration of fixed channels. The way in which channels are grouped can substantially alter the number of users handled by a trunked system. From Table 3.4, for GOS= trunked channels can support 4.46 Erlangs. Two 5 trunked channels can support 2x1.36=2.72 Erlang. 10 trunked channels support 64% more traffic than two 5 channel trunks do. 76

77 Wireless Information Transmission System Lab. Improving Capacity in Cellular Systems Institute of Communications Engineering National Sun Yat-sen University

78 System Expansion Techniques Adding New Channels Frequency borrowing Cell Splitting Sectoring / Sectorization Change of Cell Pattern Coverage zone 78

79 Cell Splitting Cell splitting is the process of subdividing a congested cell into smaller cells, each with its own base station and a corresponding reduction in antenna height and transmitter power. Cell splitting increases the capacity of a cellular system since it increases the number of times that channels are reused. 79

80 Cell Splitting Cell splitting small cells (microcells) Same service area More cells in the service area, more capacity. 80

81 Cell Splitting Cell Splitting (Hot Spot) 81

82 Transmit Power for Split cell The transmit power of the split cell must be reduced. For example, if new cell radius is half of that of old cell and the path loss exponent n = 4: P [at old cell boundary] P 1 R P r r[ at new cell boundary] Pt 2 t n ( R 2) P r[ at new cell boundary] = Pr [at old cell boundary] Pt 1 P t 2 = 16 n 82

83 Sectoring The technique for decreasing co-channel interference and thus increasing system capacity by using directional antennas is called sectoring. The factor by which the co-channel interference is reduced depends on the amount of sectoring used. # of antenna, # of handover, trunking efficiency 83

84 Sectoring 84

85 Sectoring 85

86 Chang of cell Pattern N=7, C T /7 channels per cell N=3, C T /3 channels per cell high SIR low SIR ( by-product) 86

87 A Novel Microcell Zone Concept 87

88 A Novel Microcell Zone Concept Any channel may be assigned to any zone. When a mobile travels from one zone to another, there is no handover. A given channel is active only in a particular zone. Thus interference is reduced and capacity is increased. Without the degradation in trunking efficiency. 88

89 Summaries of Improving Capacity in Cellular Systems While cell splitting increases the number of base stations in order to increase capacity, sectoring and zone microcells rely on base station antenna placements to improve capacity by reducing co-channel interference. Cell splitting and zone microcell techniques do not suffer the trunking inefficiencies experienced by sectored cells, and enable the base station to oversee all handoff chores related to the microcells, thus reducing the computational load at the MSC. 89

90 Wireless Information Transmission System Lab. Trunking Theory Institute of Communications Engineering National Sun Yat-sen University

91 Introduction There are two major classes of trunked radio systems: Lost Call Cleared (LCC) Lost Call Delayed (LCD) Lost Call Cleared system Queueing is not provided for call requests. When a user requests service, the user is given immediate access to a channel if one is available. If no new channels are available, the call is blocked without access to the system. The user is free to try the call again later. Calls are assumed to arrive with a Poisson distribution. The time between successive calls is exponentially distributed. The Erlang B formula describes the grade of service (GOS) as the probability that an user will experience a blocked call. 91

92 Introduction Assumed that there are infinite number of users. Assume that all blocked calls are instantly returned to an infinite user pool, and may be retried at any time in the future. Lost Call Delayed system: Queues are used to hold call requests that are initially blocked. When a channel is not immediately available, the call request may be delayed until a channel becomes available. The Erlang C formula gives the likelihood that all channels are in use. It is also necessary to know the probability of how long the cal must be delayed before a channel is available. The GOS is measured by the probability that calls will have delayed greater than t seconds. Assume that all calls in the queue are eventually serviced. 92

93 Wireless Information Transmission System Lab. Erlang B Formula Institute of Communications Engineering National Sun Yat-sen University

94 Basic Assumptions Call requests are memoryless, implying that all users, including blocked users, may request a channel at any time. The probability of a user occupying a channel (call the service time) is exponentially distributed. Traffic requests are described by a Poisson distribution which implies exponentially distributed call inter-arrival times. Inter-arrival times of call requests are independent of each other. There infinite number of users and there are finite number of channels available in the trunked pool. 94

95 Derivation of Erlang B Consider a system with C channels and U users. Let λ be the total mean call arrival rate per unit time for the entire trunked system (average number of call requests per unit time over all channels and all users). Let H be the average call holding time (average call duration). If A is the offered load for the trunked system, A=λH. The probability that a call requested by a user will be blocked is given by: Pr[Blocking]=Pr[None of the C channels are free] Calls arrive according to the Poisson distribution: λτ e n Pr { a( t+ τ) a( t) = n} = ( λτ) for n = 0, 1, 2... n! 95

96 Derivation of Erlang B a(t) is the number of call requests (arrivals) that have occurred since t = 0. The Poisson process implies that the time of the nth call arrival and the interarrival times between successive call requests are mutually independent. The interarrival times between call requests are exponentially distributed and mutually independent, and the probability that the interarrival time will be less than some time s is given by Pr(τ n s)=1-e -λs, s 0 where τ n is the interarrival time of the nth arrival and τ n =t n+1 -t n, where t n is the time at which the nth call request arrived. The probability density function forτ n is: n p τ = λe λτ τ ( ), 0. n 96 n

97 Derivation of Erlang B For every t 0 andδ 0 { a( t+ δ ) a( t) = } = λδ + O( δ ) { a( t+ δ) a( t) = } = λδ + O( δ) a( t+ δ) a( t) = O δ Pr 0 1 Pr 1 { } ( ) Pr 2 (A) where O(δ) is the probability of more than one call request arriving over the time interval δ and is a function ofδ such that O ( δ ) lim = 0 δ 0 δ The probability of n arrivals in δseconds is given by: λδ e Pr{ ( ) ( ) } n a t+ δ a t = n = ( λδ ) n! 97

98 Derivation of Erlang B The user service time is the duration of a particular call that has successfully accessed the trunked system. Service times are assumed to be exponentially distributed with mean call duration H, where μ=1/h is the mean service rate. The probability density function of the service time is sn p( sn ) = μe μ where s n is the service time of the nth call. This trunking system is called an M/M/C/C queueing system. The first M denotes a memoryless Poisson process for call arrivals, the 2nd M denotes an exponentially distributed service time, the first C denotes the number of channels available, and the last C indicates a hard limit on the number of simultaneous users that are served. 98

99 Derivation of Erlang B Consider a discrete time stochastic process {X n n=0, 1, 2, } that takes values from the set of nonnegative integers, so that the possible states of the process are i=0,1, 2,. The process is said to be a Markov chain if its transition from the present state i to the next i+1 depends solely on the state i and not on previous states. At time kδ, the number of calls (occupied channels) N k in the system may be represented as N k = N(kδ), where N is a discrete random process representing the number of occupied channels. The transition probability P i,j, which describes the probability of channel occupancies over a small observation interval, is given by P = Pr N = j N = i { } i, j k+ 1 k 99

100 Derivation of Erlang B Using Equation (A) and letting δ 0, we obtain: P = 1 λδ + O δ oo ii, 1 i, j ( δ ) ( ) ( δ ) ( δ ) ( ) P = 1 λδ μδ + O δ i 1 ii Pii, + 1 = λδ + O i 0 P = μδ + O i 1 P = O j i, j i+ 1, j i 1 100

101 Derivation of Erlang B At steady state, we have the Global Balance Equation: As a result, we have: λp = Pnμ n 1 n λδp n P n C = n 1 μδ n, λp 1 λ 1 λ P = P = P = P μ 2 μ 2 μ n C n= 0 P n = 1 n C C λ 1 μ λ 1 Pn = P0 P0 = Pn n! = 1 Pi = 1 P 0 μ n! λ i= 1 i= 1 μ i! 1 P0 = C n λ 1 n= 0 μ n! i

102 102 Derivation of Erlang B The probability of blocking for C trunked channels is: The total offered traffic is A=λH=λ/μ. The Erlang B formula which gives the probability of blocking can be obtained by: 0 0 1! 1! 1! C C c n C n C P P C n λ μ λ μ λ μ = = = 0 1! 1! C c C n n A C P A n = =

103 Wireless Information Transmission System Lab. Erlang C Formula Institute of Communications Engineering National Sun Yat-sen University

104 Derivation of Erlang C Assumption: if an offered call can't be assigned a channel, it is placed in a queue which has an infinite length. The Erlang C formula is derived by assuming that the trunked system is a M/M/C/D queue, where C denotes the maximum number of simultaneous users and D is the maximum number of calls that may be held in the queue for service. 104

105 Derivation of Erlang C From the state diagram: λ 1 λδ Pk 1 = kμδ Pk for k C Pk = Pk 1 for k C μ k λ 1 λδ Pk 1 = Cμδ Pk for k C Pk = Pk 1 for k C μ C P k k λ 1 P0 k C μ k! = k 1 λ 1 P k C 0 k C C! μ C 105

106 Since P k = 0 Derivation of Erlang C P k =1, we have: C+ 1 λ 1 λ = 1 ( C ) μ C! μ C C k k C 1 λ 1 1 λ 1 P = 1 k C k= 1 μ k! k= c C! μ C 1 P0 = k C C 1 λ 1 1 λ 1 k = 1 μ + k! k! μ λ 1 μc 106

107 Derivation of Erlang C r [ channels are busy] P C = k= C P k 1 λ 1 = k= CC! μ C C k k C 1 λ λ 1 = P0 C! μ k= C μ C C 1 λ 1 λ = P0 < 1 C! μ λ μ C 1 μc P 0 k C k C 107

108 Derivation of Erlang C r Substituting for P 0 [ channels are busy] P C C 1 λ C! μ 1 = λ 1 C 1 k C μc λ 1 1 λ 1 + k= 0 μ k! k! μ λ 1 μc λ μ = C k C 1 λ λ λ 1 C!1 μ + μc k= 0 μ k! C 108

109 Derivation of Erlang C With A=λ/μ=λH, the Erlang C formula is given by: [ C ] = 1 Pr channels are busy A C A C + C!1 A C C k = 0 k A k! 109