In this paper a procedure to approximately calculate the mean waiting time in the M/H2/s queue is presented. The approximation is heuristic although based in the intuitive symmetry between the deterministic and balanced hyperexponential-2 distributions. The three parameters which fully describe the H2 distribution are considered, so the approximation can also be used for the M/G/s queue when the first three moments are known. If only the first two moments of the holding time distribution are known, the estimation can also be applied accepting a lesser accuracy. The estimation proposed is a closed formula extremely easy to compute and the results are very accurate. This features makes it helpful in the design of mobile telecommunication systems with more than one channel and queueing allowed (like trunking Private Mobile Radio PMR systems), where holding time distributions with coefficients of variation higher than one may appear.
As a second stage, the possibility of calls owning a certain level of priority is studied. Two service classes are considered according to a non-preemtive priority scheme (also known as Head Of the Line or HOL). This priority feature is often required in mobile telecommunications systems to improve the access delay of some special calls by degrading the delay suffered by the rest. If the proportion of calls owning priority is kept low, the degradation is shared by many calls and then kept small. In this paper a procedure to estimate the mean waiting time in queue for each priority class is presented. This procedure is also very easy to compute.
The environment for which the results of this paper are intended suggests medium or heavy overall load and light priority load (priority proportion is kept low). This is the situation under which the accuracy of the proposed method is checked. Although simulations are necessary in the final phase of the design, the procedure presented here is helpful as a first quick insight into the system performance.Peer ReviewedPostprint (published version

Aguilar Igartua, Mónica

Barceló Arroyo, Francisco

Paradells Aspas, Josep

English

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15th IMACS World Congress EMS, Vol 4, Berlin, August 1997577Mean Waiting Time in the M/H2/s Queue: Application to MobileCommunications SystemsFrancisco Barceló, Josep Paradells, Mónica AguilarE.T.S. Ingenieros de Telecomunicación de Barcelona (U.P.C.)c/ Jordi Girona 1-3,   Mod. C3,  Barcelona 08034e-mail: barcelo@mat.upc.esKeywords: Priority queue, M/G/s queue, Queueing theory.ABSTRACTIn this paper a procedure to approximately calculate the mean waiting time in the M/H2/s queue ispresented. The approximation is heuristic although based in the intuitive symmetry between thedeterministic and balanced hyperexponential-2 distributions. The three parameters which fullydescribe the H2 distribution are considered, so the approximation can also be used for the M/G/squeue when the first three moments are known. If only the first two moments of the holding timedistribution are known, the estimation can also be applied accepting a lesser accuracy. Theestimation proposed is a closed formula extremely easy to compute and the results are veryaccurate. This features makes it helpful in the design of mobile telecommunication systems withmore than one channel and queueing allowed (like trunking Private Mobile Radio PMR systems),where holding time distributions with coefficients of variation higher than one may appear.As a second stage, the possibility of calls owning a certain level of priority is studied. Two serviceclasses are considered according to a non-preemtive priority scheme (also known as Head Of theLine or HOL). This priority feature is often required in mobile telecommunications systems toimprove the access delay of some special calls by degrading the delay suffered by the rest. If theproportion of calls owning priority is kept low, the degradation is shared by many calls and thenkept small. In this paper a procedure to estimate the mean waiting time in queue for each priorityclass is presented. This procedure is also very easy to compute.The environment for which the results of this paper are intended suggests medium or heavy overallload and light priority load (priority proportion is kept low). This is the situation under which theaccuracy of the proposed method is checked. Although simulations are necessary in the final phaseof the design, the procedure presented here is helpful as a first quick insight into the systemperformance.15th IMACS World Congress EMS, Vol 4, Berlin, August 1997578Mean Waiting Time in the M/H2/s Queue: Application to MobileCommunications SystemsFrancisco Barceló, Josep Paradells, Mónica AguilarE.T.S. Ingenieros de Telecomunicación de Barcelona (U.P.C.)c/ Jordi Girona 1-3,   Mod. C3,  Barcelona 08034e-mail: barcelo@mat.upc.esKeywords: Priority queue, M/G/s queue, Queueing theory.ABSTRACTIn this paper a procedure to approximately calculate the mean waiting time in the M/H2/s queue ispresented. The approximation is heuristic although based in the intuitive symmetry between thedeterministic and balanced hyperexponential-2 distributions. The three parameters which fullydescribe the H2 distribution are considered, so the approximation can also be used for the M/G/squeue when the first three moments are known. This paper introduces a slight modification thatimproves the approximation given in [1]. Two service classes are considered according to a non-preemtive priority scheme (also known as Head Of the Line or HOL). This priority feature is oftenrequired in mobile telecommunications systems. The situation of medium or heavy load underwhich the accuracy of the proposed method is checked is the natural condition of evaluation forthese systems: under light load the system must always work correctly. A way to estimate themean waiting time or access delay for both type of calls is presented in this paper as an extensionof [2, 3].INTRODUCTIONThe M/G/s queue is very useful to model mobile telecommunications systems because the generalservice time allows the designers to include the measured distribution of the channel holding timein the computation of the system Grade of Service (GoS). The distribution of the arrival process isin most cases more difficult to measure, and the Poisson arrival process is a reasonableassumption. As the exact solution for the M/G/s queue does not exist the designer must rely oncomputer simulations to estimate the system GoS. For the M/H2/s queue an exact solution exists[4] which can be applied to estimate the delay in the M/G/s model when the squared coefficient ofvariation of the service time distribution is higher than 1; but the degree of complexity to computethis solution is extremely high and it could be quicker to simulate the system than to obtain thementioned exact solution. This paper is concerned with systems in which the service time isdistributed with a coefficient of variation higher than one, giving an approximate formula toestimate the mean waiting time in queue (or the mean queue length related to the waiting time byLittle’s formula). The proposed approximation is extremely easy to compute and very precise.Although the results obtained by using the method presented must be completed with simulationresults in the final phase of the system design, the approximation is very helpful as a first quickinsight into the system performance.In telecommunication systems, service time distributions with coefficient of variation higher thanone are often found. Squared coefficients of variation as high as 4.8 are common in fix telephony[5]. When voice is multiplexed at talk-spurt level the squared coefficient of variation reaches 2.5[6]. In Private Mobile Radio (PMR) systems which integrate dispatch calls (with a typical durationof 20 seconds) and interconnection to public telephone network (120 seconds), the coefficient ofvariation of the service time can be any, depending on the proportions of the call mixture. Thesame can be said about the combination or integration of voice (dispatch or interconnection) and15th IMACS World Congress EMS, Vol 4, Berlin, August 1997579data calls (much shorter) in the same PMR system. Some of these systems can be evaluatedaccording to a ‘Blocked Calls Lost’ basis (no queue) and are obviously out of the scope of thispaper, but others, primarily PMR systems, should be evaluated under the ‘Blocked Calls Delayed’basis and the results for the M/G/s queue apply. If the mentioned systems are evaluated as a M/M/squeueing model which assumes a holding time distribution less disperse than the actual one, theGoS (mean access delay) is underestimated and the system tends to be undersized.The priority feature is widely spread in PMR systems: it is found in trunking systems likeMPT1327 and TETRA. We consider only the non-preemptive priority or Head Of the Line (HOL)case: if the priority call arrives when all the channels are busy, it does not interrupt a call inprogress but it is placed in the queue before the non-priority calls. The emergency calls also foundin the above mentioned systems are of the preemptive type: a priority call interrupts a non-prioritycall in progress when there is no channel available. Only priority (HOL) is considered in this paper,being emergency calls ignored.NOTATION AND REVIEW OF PREVIOUS RESULTSThe following notation is used in the paper:A = offered traffic,( ) ( )b t a t a t= − + − −µ µ µ µ1 1 2 21e e = service hyperexponential-2 p.d.function.( )1 11 2µ µ µ= + −a a = mean holding time,ρ = A s  = offered load,mi = i th ordinary moment (m1 = 1/µ  = mean),c = coefficient of variation,k mmc= =+2122212= relative 2nd moment,W M G s( / / ) = mean waiting time in the queue M/G/s,( )( )RW sW sG =M GM M= relative mean waiting time,ra= µ µ1 = proportion of time serving short-type calls.The three values chosen in this paper to describe the service time are m1, c and r. In the case of H2service time they fully describe the holding time distribution. In the general case (M/G/s) the threefirst moments of the holding time distribution can easily be related to m1, c and r. Note that thedefinition of r implies µ1 ≥ µ2.Due to the excessive complexity of the exact solution for the M/H2/s queue, many approximationsare described in the literature for the mean waiting time in the queue (see [8] for a very completelist of references). In the environment that we consider with medium or heavy load theapproximation introduced by Boxma, Cohen and Huffles [7] takes into account the first threemoments and gives excellent results while the coefficient of variation of the holding time keepsreasonably low (i.e. c2 < 3). The drawback of this approximation is that the computation iscumbersome and must be programmed.15th IMACS World Congress EMS, Vol 4, Berlin, August 1997580The approximation proposed by Kimura [8] is much simpler to compute but only takes intoaccount the first two moments of the service time distribution: it does not depend on r. This factmakes it very precise and helpful when the service time is near to be balanced (r ≅ 0.5) but theapproximation is inaccurate for other values of r. This approximation is used below in this paperand states that the relative mean waiting time can be estimated as: ( )RkRc R cGDD=+ −22 12 2(1)where RD stands for the relative mean waiting time of the M/D/s queue.PROPOSED APPROXIMATION FOR THE MEAN WAITING TIMEWhen c is high and assuming that the first three moments of the service distribution are known, theapproximation given by Boxma is more precise than Eq. (1), but much harder to compute. Forvalues of c2 > 3 both approximations start to be inaccurate. In this case the approximation given in[1] provides an excellent trade-off between accuracy and simplicity as it is extremely easy tocompute. Here a slight modification to the formula given in [1] is proposed which improves theaccuracy of the approximation. The proposed estimated value of the relative mean waiting time inqueue is:( ) ( )( )( )( )( )( )( )( )R r kk s ss k k s srH = −− − − + −+ − − − + −12 05 1 1 4 5 216 1 1 1 4 5 2. ρρ ρ(2)In Table I some numerical results of RH(0.5) are shown for different values of load, coefficient ofvariation and number of channels; Kimura approximation of Eq. (1) is used for comparison. InFigure 1 the proposed formula is compared with simulation results and the approximationproposed by Boxma for two different values of r = 0.25 and r = 0.75.Table I: Numerical results for the relative mean waiting time RH in the M/H2/s queueTHE M/H2/s QUEUE WITH PRIORITYIt is common in mobile telecommunication systems the use of priority schemes to improve theperformance for some special calls. Obviously this improvement will cause a longer access delayto regular calls, but if the proportion of priority calls is kept small, the longer access delay is sharedamong many calls and the degradation is insignificant for practical purposes. Here an easy way tos 5 12 20Load Exact Kim. New Exact Kim. New Exact Kim. New0.6 1.17 1.19 1.19c2=1.5 0.7 1.20 1.22 1.21 1.16 1.17 1.17 1.13 1.15 1.140.8 1.22 1.23 1.23 1.19 1.20 1.20 1.17 1.19 1.190.9 1.23 1.24 1.24 1.22 1.23 1.22 1.21 1.22 1.220.6 1.51 1.52 1.54c2=2.5 0.7 1.58 1.59 1.61 1.44 1.47 1.49 1.35 1.39 1.400.8 1.64 1.65 1.67 1.55 1.56 1.59 1.48 1.50 1.530.9 1.70 1.70 1.71 1.65 1.66 1.67 1.62 1.63 1.650.6 1.97 1.91 2.02c2=4 0.7 2.12 2.09 2.17 1.84 1.81 1.91 1.66 1.62 1.740.8 2.26 2.23 2.30 2.06 2.02 2.11 1.92 1.89 1.990.9 2.39 2.37 2.40 2.29 2.25 2.31 2.21 2.17 2.2415th IMACS World Congress EMS, Vol 4, Berlin, August 1997581estimate the mean waiting time is presented as an extension of [2, 3] where the deterministicservice time and coefficients of variation lower than one are considered.1,822,22,42,62,832 4 6 8 10 12 14 16 18 20 22 24 26 28 30sRHBox-0.25Prop-0.25Box-0.75Prop-0.75Sim-0.25Sim.-0.75Figure 1: Comparative results for load = 85% and c2=5. The proportions of time spent in short-type calls are r = 0.25 and r = 0.75.To estimate the access delay for each type of calls (priority and non-priority) the following stepsmust be followed (index 1 stands for priority calls and 2 for non-priority):Step1: The access delay for priority calls should be calculated according to Eq. (1). It isassumed that the priority proportion p is low so Eq. (2) is not valid as it is heuristically based in anapproximation which is inconsistent for low load (see [1, 2, 3] for further details). The value of RDin Eq. (1) must be calculated according to the linear approximation which is not so accurate as thebelow mentioned Eq. (5) but keeps its validity for low loads (is asymptotically exact when ρ → 0):( )RW M D sW M M sp sspD11111 2= =−++( / / )( / / )ρ ρ(3)The mean waiting time for the priority calls in the M/M/s equivalent queue can be computedaccording to the exact value given in [9]:( ) ( )W M M s Erlang c A s s 1 p1( / / ) ,= − −1 µρ(4)Step 2: The access delay for all calls can be calculated according to Eq. (2) as the overall loadis heavy or medium. RD should now be calculated according to the following excellentapproximation:( )RW M D sW M M ss ssD= = +− − + −( / / )( / / )( )( )1211 1 4 5 216ρρ (5)and W M M s( / / ) can be calculated according to the Erlang-c formula.Step 3: The mean access time for non-priority calls can be computed from the followingaverage waiting time:( )W M H s pW M H s p W M H s( / / ) ( / / ) ( / / )2 1 2 2 21= + − (6)where the only unknown is the mean waiting time for non-priority calls.15th IMACS World Congress EMS, Vol 4, Berlin, August 1997582NUMERICAL RESULTSThe proposed procedure can only be validated by numerical tests, comparing simulation resultswith the values obtained using steps 1 to 3 of the previous section; to this purpose extensivenumerical tests have been performed under different conditions of load (overall and priority),number of channels, coefficient of variation and third moment (described through r). In Figure 2 asystem consisting of 5 channels 90% loaded is assumed with priority proportion p varying between0 and 0.5 and average call duration normalised to one time unit. Results of the mean waiting timeobtained by simulation and by using the procedure proposed here to approximate the mean waitingtime are plotted for both priority (Figure 1a) and non-priority calls (Figure 2b). The valuescalculated according to the Erlang-c formula for the M/M/s queue are also plotted for reference.0.150.200.250.300.350.400.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50pDelay WM1 Appr. 1 Sim. 11.002.003.004.005.006.000.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50pDelay WM2 Appr. 2 Sim. 2Figure 2: Mean access delay in a system with 5 channels 90% loaded for c2=3a) Priority calls ; b) Regular callsIn table II values of the mean waiting time for priority calls obtained by simulation are shownalong with the values obtained by using the proposed approximation for different values of thesquared coefficient of variation, load, priority proportion and number of available channels.Table II: Mean access delay for priority calls; mean call duration normalised to 1,000 time units.C 5 1 2 2 0c2 Load p Appr. Sim. Appr. Sim. Appr. Sim.0.90 0.1 176 179 60 61 31 311.5 0.95 0.1 204 205 76 76 42 420.90 0.2 198 201 67 67 35 350.95 0.2 230 230 87 87 48 480.90 0.1 187 191 62 63 32 322.5 0.95 0.1 217 221 79 80 44 440.90 0.2 213 216 71 71 36 360.95 0.2 249 258 90 90 50 500.90 0.1 196 214 64 64 32 334.0 0.95 0.1 228 252 81 83 44 440.90 0.2 226 253 73 74 37 380.95 0.2 264 294 94 95 51 51In Figure 3 the dependency of the mean waiting time for priority and non-priority calls with respectto the squared coefficient of variation of the holding time distribution is plotted. A system with 7channels loaded 85% with 10% of priority calls is considered.15th IMACS World Congress EMS, Vol 4, Berlin, August 19975830.00.51.01.52.02.53.01 2 3 4 5 6 7 8 9 10c2Delay W1 (Appr.) x5 W1 (Sim.) x5 W2 (Appr.) W2 (Sim.)Figure 3: Mean access delay versus squared coefficient of variationCONCLUSIONSDue to the complexity of the exact solution for the waiting time distribution in the M/H2/s queue,approximated results are helpful when they are accurate and easy to compute. In this paper a closedformula to estimate the mean waiting time in queue is given. The approximation is checked in theenvironment of mobile communications (medium or heavy load and multi-channel) proving to beprecise. A procedure to estimate the access delay in the case that the system features some priorityscheme is also presented. This latter estimation is also precise and simple to compute. All theresults presented are useful for quick evaluation of communication systems with a model morerealistic than the conventional M/M/s.REFERENCES[1] F. Barceló, J. Paradells, The M/H2/s queue in mobile communications: Approximation of themean waiting time, 14th U.K. Teletraffic Symposium, IEE (1997).[2] F. Barceló, V. Casares, J. Paradells, The M/D/C Queue with Priority: Application to trunkedMobile Radio Systems  IEE Electronics Letters, Vol 32, No. 18 , p. 1644 (1996).[3] F. Barceló, J. Paradells, Performance Evaluation of Public Access Mobile Radio (PAMR)Systems with Priority Calls, 5th Int. Conf. on Telecommunication Systems, p. 360 (1997).[4] J. H. A. De Smit, A Numerical Solution for the Multi-Server Queue with Hyper-ExponentialService Times, Operations Research Letters, Vol. 2, No. 5, p. 217 (1983).[5] V. Bolotin, Telephone Circuit Holding Time Distributions,  Proc. 14th Inernational TeletrafficCongress,  p. 125 (1994).[6] H. H. Lee, C.K. Un, A study of On-Off Characteristics of Conversational Speech, IEEE Trans.on Communications, COM-34, num. 6, p. 630 (1986).[7] O.J. Boxma, J.W. Cohen, N. Huffles, Approximations of the Mean Waiting Time in an M/G/sQueueing System, Operations Research, Vol. 27, p. 1115 (1979).[8] T. Kimura, Approximations for multi-server queues: system interpolation, Queuing Systems17, p. 347 (1994).[9] D. Gross, C. M. Harris, Fundamentals of Queueing Theory, John Wiley & Sons (1974).

Mean waiting time in the M/H2/s queue: application to mobile communications Systems

In this paper a procedure to approximately calculate the mean waiting time in the M/H2/s queue is presented. The approximation is heuristic although based in the intuitive symmetry between the deterministic and balanced hyperexponential-2 distributions. The three parameters which fully describe the H2 distribution are considered, so the approximation can also be used for the M/G/s queue when the first three moments are known. If only the first two moments of the holding time distribution are known, the estimation can also be applied accepting a lesser accuracy. The estimation proposed is a closed formula extremely easy to compute and the results are very accurate. This features makes it helpful in the design of mobile telecommunication systems with more than one channel and queueing allowed (like trunking Private Mobile Radio PMR systems), where holding time distributions with coefficients of variation higher than one may appear.

As a second stage, the possibility of calls owning a certain level of priority is studied. Two service classes are considered according to a non-preemtive priority scheme (also known as Head Of the Line or HOL). This priority feature is often required in mobile telecommunications systems to improve the access delay of some special calls by degrading the delay suffered by the rest. If the proportion of calls owning priority is kept low, the degradation is shared by many calls and then kept small. In this paper a procedure to estimate the mean waiting time in queue for each priority class is presented. This procedure is also very easy to compute.

The environment for which the results of this paper are intended suggests medium or heavy overall load and light priority load (priority proportion is kept low). This is the situation under which the accuracy of the proposed method is checked. Although simulations are necessary in the final phase of the design, the procedure presented here is helpful as a first quick insight into the system performance.Peer Reviewe

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Mean waiting time in the M/H2/s queue: application to mobile communications Systems

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