Mean Waiting Time Analysis in Finite Storage Queues for Wireless Cellular Networks

Transcription

1 Mean Waiting Time Analysis in Finite Storage ueues for Wireless ellular Networks J. YLARINOS, S. LOUVROS, K. IOANNOU, A. IOANNOU 3 A.GARMIS 2 and S.KOTSOOULOS Wireless Telecommunication Laboratory, Department of Electrical and omputer Engineering, University of atras, Rion 265, atras 2 Department of Applied Informatics in Management & Economy, Tecnological Educational Institution of Messolongi, Messolongi, 3 3 ellenic Telecommunications Organization S.A. Marousi, 99 Kifissias Ave. ostal ode 5 24 Atens Abstract: - uality of service is obtained by introducing a tresold in te maximum waiting time of a andover call in te queue. In case te andover call mean service time at eac queue position is found to be greater tan tis tresold, tis call can not be serviced by tis cell. Tis innovative idea introduces a more flexible traffic management since it enables a delay prediction in case of congestion. Key-Words: -Wireless Networks, os, Mean Waiting Time Introduction In case of microcellular networks were frequent andovers is a fact, uality of Service (os may degenerate below an acceptable level due to brief service interruptions. As te frequency of tese interruptions increases te perceived os is reduced. Tis issue places additional callenges on te design and dimensioning of microcellular wireless networks. Increasing te andover rate, te probability of an ongoing call to be dropped due to a lack of free cannel is ig. In order to preserve tis blocking a queue is introduced to accommodate te andover calls. owever te queue size sould be kept small and te time spent in te queue sould be below a certain tresold since andover attempt and waiting sould not last for ever. Tis tresold sould be decided after a matematical ustification over certain conditions from teory. In tis paper a matematical analysis is conducted in order to model te waiting time in te queue as function of mean service time, queue lengt and offered traffic load. 2 Matematical Analysis of te roposed andover rocedure riority can be given to andover attempts by assigning guard cannels exclusively for andover calls among te cannels in a cell. Bot te new and te andover calls can sare te remaining cannels. A andover attempt may be queued in a queue of k size if te state number in te cell is equal to (All cannels in te cell are busy. Te cannel olding time T in a cell is defined as te time duration between te instant tat a cannel is occupied by a call and te instant it is released by eiter completion of te call or a cell boundary crossing by a portable, wicever is less. It is proved [] tat te pdf of T can be approximated to a negative exponential distribution wit meant Te service timet, te time tat an attempt remains queued at a position q, depends normally

2 on weter or not a cannel becomes available as long as te mobile is still in te andover area. A andover attempt tat oins te queue will be successful, if bot of te following events occur before te mobile moves out of te andover area:. All of te attempts wic oined te queue earlier tan te given attempt ave been disposed 2. A cannel becomes available wen te given attempt is at te first position in te queue. Using te steady-state equations from Figure, we conclude: + λn λn λ + E E - E E +k ( ( k Figure : State transition diagram of te queuing traffic model!! i ρ,! ( (, ( <, < + k ( ρ ( + k + +! +! +! i ( ( were λn and are te new calls and te andover calls arrival rate respectively, λ + λ is te total call arrival rate ( λ aλ and te offered load ρ in a communication system is defined as ( λ + λn ρ. On te basis of our consideration, T sould ave an upper bound. In order to ave an effective system, a andover call must not be allowed to n remain at a buffer position more tan a maximum time tresold. Moreover, te queue size as to be limited because it is more realistic and practical tan te infinite buffering. Te maximum value of te mean service time T is obtained by te mean waiting time W in te queue. Waiting time of a queued andover call is defined as te time of an arbitrarily selected andover call between te moment it is accepted and begin waiting in te queue to te moment it successfully accesses a free

3 cannel. Given tat te state of te system is wen te call arrives and waits in te queue, we denote te waiting time by W ( [3]: W ( ln( R ( (2 were R ( is te dropping probability of an arbitrary selected andover call, given tat te system state is + q ( q k ust at te instant te call is accepted by te system and waits in te queue. onsequently, te average waiting time of a andover call, denoted byw, can be obtained by: + k W ( W + k (3 As we can easily conclude, W is, among oter, a, T ( T function of te mean service timet /. Tus, r( call _ remains _ in _ queue, oterwise setting an upper bound ( W for te waiting time (6 Substituting equations (5 and (6 into (4, in te queue, we can solve for T and find te R ( + q can be calculated and ten te mean corresponding maximum allowable mean service time at every position q. Of course, tis solution of waiting time of a andover call in queue W can be T sould be inside te interval [, +. estimated as a function oft. Setting a maximum As we already mentioned, te blocked andover value for W (after certain logical calls oin a queue. A andover attempt tat enters te queue at te position q will be successful, if it constraints, T can be calculated as a reverse manages to reac te first position of te queue and function of W. get a cannel before its mean service time becomes greater tan te calculated from equation (3 value. In order to derive te probability of a andover failure in te queue R ( + q, we assume tat: 5. Simulation Results Te following assumptions ave been made q R ( + q ( i / i + r( call _ remains _ in _ during queuesimulation: i Te total number of available cannels in te (4 cell is 2. Te probability of transition from position i + to 2 cannels are reserved only for i is denoted by ( i / i + in equation (4 and is andover calls. contributed by two probabilities [2]: I. Te remaining cannel olding time of any of Te andover call to total call is a. Tis 3 te calls in progress is smaller tan eac of value is based on statistical measurements in te following: real cellular systems. Te remaining cannel olding time of any of te oter ( calls in progress. Te service time of any of te i waiting andover calls. Te service time of te waiting andover call of interest. II. Te remaining service time of any of te i andover calls waiting in te queue is smaller tan eac of te following: Te cannel olding time of any of te calls in progress, Te service time of any of te oter ( i waiting andover calls. Te service time of te waiting andover call of interest. Tus, te transition probability can be obtained by: ( i / i + (5 + ( i + Te second term in equation (4 is a logical condition tat can ave only two values. If te mean service time at tis position is smaller tan or equal to te maximum mean service time tresold, tis term is set to. Oterwise, it is set to. Tus: Te mean cannel olding time is sec -, according to mean value 85 analysis of real cellular network data.

4 Te relation between service and waiting time is presented in figures 2 and 3. In figure 2 we notice tat, as te offered load increases, te increment inw is more significant for a queue size of k 5 tan for a size of k 2. Tis is because we accommodate more subscribers in a larger queue and te service is delayed. owever te more te andover calls to accommodate te more likely is te probability of forced terminating a andover call due to exceeding te maximum service time. In figure 3, te mean waiting time is plotted as a function of offered load wit parametric values of specific service time at eac queue position (k2 or k5. It is obvious tat te mean service time of.4 sec is too low (resulting from a very low tresold ( W and as a result te mean waiting time is increasing very slowly (around te neigborood value of sec wit offered load. owever a small increment of te mean service time up to.8 sec results in a large increment of te mean waiting time, sowing ow sensitive is te mean waiting time in small perturbations of service time. Tis as to be taken into consideration in case of dimensioning te network. Te calculation of te maximum service time in queue is a very useful parameter to calculate te andover blocking probability. andover blocking probability is a maor criterion of os of cellular networks. ence modeling te andover blocking probability as a function of maximum service time gives a powerful tool to te designer to calculate te capacity of te system and evaluate te system response. Moreover if we consider also te queue lengt k ten te andover blocking probability will be a function of tree variables (mean service time, queue lengt and offered load, providing te designer wit a more powerful tool to design te network resources, taking care also of special compromises among te parameters.

5 Figure 2: Mean ueue Waiting Time vs. Offered Load for different queue sizes Figure 3: Mean ueue Waiting Time vs. Offered Load for different mean queue service times. References reneging/dropping, IEEE/AM Trans. on networking, vol. 2, No 2, April 994. [] ong D. and Rappaport S.S.: Traffic model and performance analysis for cellular mobile radio telepone systems wit prioritized and non prioritized andover procedures IEEE Trans. on Veicular Tecnology, vol. VT-35, pp.77-9, 985. [2] anoutsopoulos I., Kotsopoulos S., Ioannou. and Louvros S.: A new proposed priority tecnique to optimize te andover procedure in personal communication systems IEE Electronics Letters, vol. 36, No 7, pp , Marc 2. [3] ang ung-ju, Su Tlan-Tsair and iang Yue-Yiing: Analysis of a cutoff priority cellular radio system wit finite queuing and