北邮通信网第二章信源模型和MM1排队系统习题答案（定稿）

第一篇：北邮通信网第二章信源模型和MM1排队系统习题答案（定稿）

第二章 通信信源模型和M/M/1排队系统－习题答案

2-1 验证性质2-4，并且说明性质2-1和性质2-4一致。

解：两个独立的Poisson过程，参数为 1和2。根据定理2－2，两个Poisson过程的到达间隔为参数1和2的负指数分布T1,T2。下面说明混合流的到达间隔，设参数1的Poisson流为红球，参数为2的Poisson流为黑球。

不妨设这个时刻到达为黑球，则下一个黑球的到达间隔为T2，而下一个红球到达间隔为T1的残余分布，由于间隔服从负指数分布，故此残余分布于原始分布一致。所以，混合流的到达间隔服从min(T1,T2),也就是参数为12的负指数分布。

T1的原始分布T1的残余分布T2性质2－4的验证

（1）Tmin(T1,T2)是一个以12为参数的负指数分布

PTtPminT1,T2tPT1t,T2tPT1tPT2te1te2te12t（3）PT1T2|Tt

112

PT1T2|Ttlimlim

PtT1tt,T2tt0PtT1tt1e1t limt01e12tt0e12te12tt1tt1te2tee1122-2 验证M/M/1的状态变化为一个生灭过程。

解：M/M/1排队系统在有顾客到达时，在时间t,tt内从状态k转移到k+1（k>=0）的概率为tot，为状态k的出生率；

当有顾客服务完毕离去时，在时间t,tt内从状态k转移到k-1（k>=1）的概率为tot，为状态k的死亡率；

在时间t,tt内系统发生跳转的概率为ot；

在时间t,tt内系统停留在状态k的概率为1tot； 故M/M/1排队系统的状态变化为生灭过程。

2-3 对于一个概率分布pk，令gXp0p1xp2x...pkxk 称为分布

2k0pk的母函数。利用母函数求M/M/1队长的均值和方差。

解：对于M/M/1

pkk(1)

k0

g(z)(1)(1)z...(1)E[k]g'(z)/z1211z 1k1Var[k]kpk[kpk]2g''(z)/z1E[k](E[k])2k121

2-4 两个随机变量X,Y取非负整数值，并且相互独立，令Z=X+Y，证明：Z的母函数为X,Y母函数之积。根据这个性质重新证明性质2-1。

证：设X的分布为：p0,p1,p2...，Y的分布为：q0,q1,q2...由于

pZkpXYkpXr,YkrpXrpYkrprqkrr0r0r0kkk

p0p1xp2x2...q0q1xq2x2...p0q0p0q1p1q0x...p0qkp1qk1...pkq0xk...

所以 g(Z)=g(X)g(Y)

对于两个独立的Poisson流，取任意一个固定的间隔T，根据Poisson过程性质，到达k个呼叫的概率分别为：

(iT)kiTpk(T)e

i=1,2 这两个分布独立

k!分布列的母函数分别为：

(iT)kkiTiTxiTiT(x1)p(T)xxeeeekk!k0k0k他们母函数之积为合并流分布列的母函数，而母函数之积e所以

合并流为参数12的 Poisson过程。

1T(x1)2T(x1)ee(12)T(x1)

2-5 如果一个连续分布满足无记忆特性，证明它就是负指数分布。

无记忆特性：对于t,s0，有Pxts|xtPxs 证明：

Pxts|xtPxsPxtsPxsPxtPxtsPxsPxt ftsftfsftcett代入初始值f01，则c1，故Pxte

2-7 求k+1阶爱尔兰（Erlang）分布Ek1的概率密度。

(x)kxe

x>=0 可以根据归纳法验证，Ek1的概率密度为

k!证明：

利用两个随机变量的和的概率密度表达式：求ZXY的分布，当X和Y相互独立时，且边缘密度函数分别为fXx和fYy，则fZzfXxfYzxdx。

k1阶Erlang分布是指k1个彼此独立的参数为的负指数分布的和。

用归纳法。

2x当k1时，需证2阶Erlang分布的概率密度为xe f1tetxetxdx2etdxt2et

t(t)kte 令nk时成立，即fktk!则当nk1时，(x)kxtxfk1tfkxftxdxeedxk!k2k1t(t)etxkdxetk!k1!tt得证

第二篇：数据、模型与决策(运筹学)课后习题和案例答案003

CHAPTER 3 THE ART OF MODELING WITH SPREADSHEETS Review Questions 3.1-1 The long-term loan has a lower interest rate.3.1-2 The short-term loan is more flexible.They can borrow the money only in the years they need it.3.1-3 End with as large a cash balance as possible at the end of the ten years after paying off all the loans.3.2-1 Visualize where you want to finish.What should the ―answer‖ look like?

3.2-2 First, it can help clarify what formula should be entered for an output cell.Second, hand calculations help to verify the spreadsheet model.3.2-3 Sketch a layout of the spreadsheet.3.2-4 Try numbers in the changing cells for which you know what the values of the output cells should be.3.2-5 Relative references are based upon the position relative to the cell containing the formula.Absolute references refer to a specific cell address.3.3-1 Enter the data first.3.3-2 Numbers should be entered separately from formulas in data cells.3.3-3 With range names, the formulas and Solver dialogue box contain descriptive range names rather than obscure cell references.Use a range name that corresponds exactly to the label on the spreadsheet.3.3-4 Borders, shading, and colors can be used to distinguish data cells, changing cells, output cells, and target cells on a spreadsheet.3.3-5 Three.One for the left-hand-side, one for the inequality sign, and one for the right-hand-side.3.4-1 Try different values for the changing cells for which you can predict the correct result in the output cells and see if they calculate as expected.3.4-2 Control-~ on a PC(command-~ on a Mac).3.4-3 The auditing tools can be used to trace dependents or precedents for a given cell.3-1 Problems 3.1 A******81920BCDEFGHIJKLLT RateST RateSavings InterestStart BalanceMinimum CashYear*********%10%3%10.5CashFlow-8-2-4363-47-210LTLoan7.50STLoan0.002.517.275.510.5702.59000(all cash figures in millions of dollars)LTInterest-0.53-0.53-0.53-0.53-0.53-0.53-0.53-0.53-0.53-0.53STInterest0.00-0.25-0.73-0.55-0.060-0.26000LTPaybackSTPayback0.00-2.51-7.27-5.51-0.570-2.59000SavingsInterest0.0150.0150.0150.0150.0150.0709260.0150.1242340.0522110.338027MinimumBalance0.500.500.500.500.500.500.500.500.500.500.503.2

a.The COO will need to know how many of each product to produce.Thus, the decisions are how many end tables, how many coffee tables, and how many dining room tables to produce.The objective is to maximize total profit.-7.50Balance0.500.500.500.500.502.360.504.141.7411.273.58>=>=>=>=>=>=>=>=>=>=>= b.Pine wood used

Labor used

c.=(3 end tables)(8 pounds/end table)+(3 dining room tables)(80 pounds/dining room table)= 264 pounds =(3 end tables)(1 hour/end table)+(3 dining room tables)(4 hours/dining room table)= 15 hours

Coffee TablesDining Room Tables

End TablesUnit ProfitResource Used per unit ProducedPine WoodLaborEnd TablesUnits ProducedCoffee TablesDining Room TablesTotal Used<=<=AvailableTotal Profit BCDEFG d.A123456789End TablesUnit Profit$50Pine WoodLaborCoffee TablesDining Room Tables$100$220Total Used3000<=200<=Available3000200Total Profit$10,600Resource Used per unit Produced81580124Coffee TablesDining Room Tables4030End TablesUnits Produced0 3-2 3.3 a.Top management will need to know how much to produce in each quarter.Thus, the decisions are the production levels in quarters 1, 2, 3, and 4.The objective is to maximize the net profit.b.Ending inventory(Q1)

Ending inventory(Q2)

Profit from sales(Q1)Profit from sales(Q2)Inventory Cost(Q1)Inventory Cost(Q2)c.Inventory Holding CostGross Profit from SalesStartingInventoryQuarter 1Quarter 2Quarter 3Quarter 4MaximumProduction<=<=<=<=Demand/EndingSalesInventory>=>=>=>=InventoryCostGross Profitfrom Sales

= Starting Inventory(Q1)+ Production(Q1)– Sales(Q1)= 1,000 + 5,000 – 3,000 = 3,000

= Starting Inventory(Q2)+ Production(Q2)– Sales(Q2)= 3,000 + 5,000 – 4,000 = 4,000

= Sales(Q1)*($20)=(3,000)($20)= $60,000 = Sales(Q2)*($20)=(4,000)($20)= $80,000

= Ending Inventory(Q1)*($8)=(3,000)($8)= $24,000 = Ending Inventory(Q2)*($8)=(4,000)($8)= $32,000

ProductionNet Profit M d.A12345678910BCDEFGHIJKLInventory Holding CostGross Profit from SalesStartingInventory1,0000$8$20Production2,0004,000MaximumProduction<=6,000<=6,000Demand/EndingSalesInventory3,0000>=04,0000>=0InventoryCost$0$0Totals$0Net ProfitGross Profitfrom Sales$60,000$80,000$140,000$140,000Quarter 1Quarter 2 e.A***BCDEFGHIJKLMInventory Holding CostGross Profit from SalesStartingInventory1,0001,0003,0001,000$8$20Production3,0006,0006,0006,000MaximumProduction6,0006,0006,0006,000Demand/EndingSalesInventory3,0001,0004,0003,0008,0001,0007,0000InventoryCost$8,000$24,000$8,000$0Totals$40,000Net ProfitGross Profitfrom Sales$60,000$80,000$160,000$140,000$440,000$400,000Quarter 1Quarter 2Quarter 3Quarter 4<=<=<=<=>=>=>=>=0000 3.4 a.Fairwinds needs to know how much to participate in each of the three projects, and what their ending balances will be.The decisions to be made are how much to participate in each of the three projects.The objective is to maximize the ending balance at the end of the 6 years.3-3 b.Ending Balance(Y1)

Ending Balance(Y2)

c.Starting Cash= Starting Balance + Project A + Project C + Other Projects

= 10 +(100%)(–4)+(50%)(–10)+ 6 = 7(in $millions)

= Starting Balance + Project A + Project C + Other Projects = 7 +(100%)(–6)+(50%)(–7)+ 6 = 3.5(in $millions)

Year123456ParticipationCash Flow(at full participation, $million)Project AProject BProject CTotalCash FlowFrom ABCOtherProjectsEndingBalance>=>=>=>=>=>=MinimumBalance<=100%<=100%<=100%

EFGHI d.A***131415BCDStarting Cash10all cash numbers are in $millionsTotalCash FlowFrom ABC00OtherProjects66EndingBalance1622MinimumBalance>=1>=1Year12Cash Flow(at full participation, $million)Project AProject BProject C-4-8-10-6-8-7Participation0%<=100%0%<=100%0%<=100% EFGHI e.A***131415BCDStarting Cash10all cash numbers are in $millionsTotalCash FlowFrom ABC-10.75-8.125-8.125-0.5-344OtherProjects666666EndingBalance5.253.12516.59.559.5MinimumBalance111111Year123456ParticipationCash Flow(at full participation, $million)Project AProject BProject C-4-8-10-6-8-7-6-4-724-4-5030-3004418.75%<=100%0%<=100%100%<=100%>=>=>=>=>=>= 3-4 3.5 a.Web Mercantile needs to know each month how many square feet to lease and for how long.The decisions therefore are for each month how many square feet to lease for one month, for two months, for three months, etc.The objective is to minimize the overall leasing cost.b.Total Cost =(30,000 square feet)($190 per square foot)+(20,000 square feet)($100 per square foot)= $7.7 million.c.Month of Lease:1Length of Lease:1Month 1Month 2Month 3Month 4Month 5Cost of Lease(per sq.ft.)Total Cost

1213Month Covered by Lease?***123414251TotalLeased(sq.ft.)>=>=>=>=>=SpaceRequired(sq.ft.)Lease(sq.ft.)BCDEFG d.A12345678910Month Covered by Lease?TotalMonth of Lease:112LeasedLength of Lease:121(sq.ft.)Month 11130,000>=Month 21120,000>=Cost of Lease$65(per sq.ft.)$100$65SpaceRequired(sq.ft.)30,00020,000Lease(sq.ft.)10,00020,000ABCDEFGHI0JKLM33N41Total Cost$2,650,000 O42P51QTotalLeased(sq.ft.)30,00030,00040,00030,00050,000RSSpaceRequired(sq.ft.)30,00020,00040,00010,00050,000 e.***13Month of Lease:1Length of Lease:1Month 11Month 2Month 3Month 4Month 51211***1511111$190Month Covered by Lease?***11111***$1001$65>=>=>=>=>=Cost of Lease$65$100$135$160(per sq.ft.)Lease(sq.ft.)0000$65$100$135$160$65$100$135$653.6

a.Larry needs to know how many employees should work each possible shift.Therefore, the decision variables are the number of employees that work each shift.The objective is to minimize the total cost of the employees.30,000000010,000000020,000Total Cost$7,650,0003-5 b.Working 8am-noon: 3 FT morning + 3 PT = 6 Working noon-4pm: 3 FT morning + 2 FT afternoon + 3 PT = 8 Working 4pm-8pm: 2 FT afternoon + 4 FT evening + 3 PT = 9 Working 8pm-midning: 4 FT evening + 3 PT = 7 Total cost per day =(3+2+4 FT)(8 hours)($14/hour)+(12 PT)(4 hours)($12/hour)= $1,584.c.Full Time8am-4pmCost per ShiftShift Covers Time of Day?(1=yes, 0=no)8am-noonnoon-4pm4pm-8pm8pm-midnightWorkers per ShiftTotalWorking>=>=>=>=TotalNeededFull Timenoon-8pmFull Time4pm-midnightPart Time8am-noonPart Timenoon-4pmPart Time4pm-8pmPart Time8pm-midnight

TotalTime of DayFull Time8am-noonnoon-4pm4pm-8pm8pm-midnightTimes TotalPart Time>=>=>=>=TotalCost

CDEFGHIJK d.A******819BFull Time8am-4pmCost per Shift$1128am-noonnoon-4pm4pm-8pm8pm-midnightWorkers per Shift11Full Timenoon-8pm$112Full Time4pm-midnight$112Part Time8am-noon$48Part Timenoon-4pm$48Part Time4pm-8pm$48Part Time8pm-midnight$48TotalWorking68126>=>=>=>=TotalNeeded68126112Shift Covers Time of Day?(1=yes, 0=no)1111162Times TotalPart Time448022410TotalCost$1,72843.7

a.Al will need to know how much to invest in each possible investment each year.Thus, the decisions are how much to invest in investment A in year 1, 2, 3, and 4;how much to invest in B in year 1, 2, and 3;how much to invest in C in year 2;and how much to invest in D in year 5.The objective is to accumulate the maximum amount of money by the beginning of year 6.TotalTime of DayFull Time8am-noon4noon-4pm64pm-8pm88pm-midnight6>=>=>=>=3-6 b.Ending Cash(Y1)= $60,000(Starting Balance)– $20,000(A in Y1)= $40,000 Ending Cash(Y2)= $40,000(Starting Balance)– $20,000(B in Y2)

– $20,000(C in Y2)= $0 Ending Cash(Y3)= $0(Starting Balance)+ $20,000(1.4)(for investment A)= $28,000 Ending Cash(Y4)= $28,000(Starting Balance)Ending Cash(Y5)= $28,000(Starting Balance)+ $20,000(1.7)(investment B)

= $62,000 Ending Cash(Y6)= $62,000(Starting Balance)+ $20,000(1.9)(investment C)

= $100,000 c.Beginning BalanceMinimum BalanceInvestmentAYear1Year 1Year 2Year 3Year 4Year 5Year 6Dollars InvestedA2A3A4B1B2B3C2D5EndingBalance>=>=>=>=>=>=MinimumBalance

BCDEFGHIJK d.A12345678910Beginning Balance$60,000Minimum Balance$0InvestmentYearYear 1Year 2Year 3A1-11.4$0A2-1-1$0$0$0A3B1-1B2-1-1$0$0B3C2-1EndingBalance$0$0$84,000MinimumBalance>=$0>=$0>=$0Dollars Invested$60,000 JKLM e.A***13BCDEFGHIBeginning Balance$60,000Minimum Balance$0InvestmentYearYear 1Year 2Year 3Year 4Year 5Year 6A1-11.41.41.41.4$0$84,000$0$0$0A2-1-1-11.71.71.7$01.9$0-11.3$117,600A3A4B1-1B2-1-1B3C2-1D5EndingBalance$0$0$0$0$0$152,880MinimumBalance$0$0$0$0$0$0>=>=>=>=>=>=Dollars Invested$60,000 3-7 3.8 In the poor formulation, the data are not separated from the formula—they are buried inside the equations in column C.In contrast, the spreadsheet in Figure 3.6 separates all of the data in their own cells, and then the formulas for hours used and total profit refer to these data cells.In the poor formulation, no range names are used.The spreadsheet in Figure 3.6 uses range names for UnitProfit, HoursUsed, TotalProfit, etc.The poor formulation uses no borders, shading, or colors to distinguish between cell types.The spreadsheet in Figure 3.6 uses borders and shading to distinguish the data cells, changing cells, and target cell.The poor formulation does not show the entire model on the spreadsheet.There is no indication of the constraints on the spreadsheet(they are only displayed in the Solver dialogue box).Furthermore, the right-hand-sides of the constraints are not on the spreadsheet, but buried in the Solver dialogue box.The spreadsheet in Figure 3.6 shows all of the constraints of the model in three adjacent cells on the spreadsheet.Cell F16 has –0.47 for LT Interest, rather than –LTRate*LTLoan.Cell G14 for the 2006 ST Interest uses the LT Loan amount rather than the ST Loan amount.Cell H21 for LT Payback refers to the 2006 ST Loan rather than the LT Loan to determine the payback amount.Cell G21 for the 2013 ST Interest uses LTRate instead of STRate.Cell H21 for LT Payback in 2013 as –6.649 instead of –LTLoan.Cell I15 for ST Payback in 2007 has –LTLoan instead of –E14(LT Loan for 2006).3.9 3.10 Case 3.1 a.PFS needs to know how many units of each of the four bonds to purchase, how much to invest in the money market, and their ending balance in the money market fund each year after paying the pensions.The decisions are how many units of each bond to purchase, as well as the initial investment in 2003 in the money market.The objective is to minimize the overall initial investment necessary in 2003 in order to meet the pension payments through 2012.3-8 b.Payment received from Bond 1(2004)=(10 thousand units)($1,000 face value)+(10,000 units)($1,000 face value)(0.04 coupon rate)= $10.4 million Payment received from Bond 1(2005)= $0

Payment received from Bond 2(2004)=(10 thousand units)($1,000 face value)(0.02 coupon)= $0.2 million Payment received from Bond 2(2005)=(10 thousand units)($1,000 face value)(0.02 coupon)= $0.2 million

Balance in money market fund(2003)= $28 million(initial investment)

– $8 million(pension payment)= $20 million

Balance in money market fund(2004)= $20 million(starting balance)

+ $10.4 million(payment from Bond 1)

+ $0.2 million(payment from Bond 2)

– $12 million(pension payment)

+ $1 million(money market interest)= $19.6 million Balance in money market fund(2005)= $19.6 million(starting balance)

+ $0.2 million(payment from Bond 2)

– $13 million(pension payment)

+ $0.98 million(money market interest)= $7.78 million

c.PFS will need to track the flow of cash from bond investments, the initial investment, the required pension payments, interest from the money market, and the money market balance.The decisions are the number of units to purchase of each bond.Data for the problem include the yearly cash flows from the bonds(per unit purchased), the money market rate, and the minimum required balance in the money market fund at the end of each year.A sketch of a spreadsheet model might appear as follows.Money Market RateMinimum Required BalanceRequiredPensionFlowMoneyMarketInterestMoneyMarketBalance>=>=>=>=>=>=>=>=>=>=0000000000

Bond Cash Flows(per unit)Bond 1Bond 2Bond 3Bond ******01020112012Units PurchasedBondFlowInitialInvestment

3-9 d.The bond cash flows(per unit)are calculated in B7:E9.For example, one unit of Bond 1 costs $0.98 in 2003, and returns the face value($1)plus the coupon rate($0.04)in 2004.The total cash flow from bonds is then calculated in column F.The Initial Investment(G7)is both a decision variable and the target cell.It includes all money invested on January 1, 2003(including enough to pay for the bonds and pension payment in 2003, as well as any initial investment in the money market).If just years 2003 through 2005 are considered, then 23.44 thousand units of Bond 1 should be purchased at a cost of $22.97 million, along with an initial $8 million investment in the money market fund on January 1, 2003.A***1314BCDEFGHIJKLMoney Market RateMinimum Required BalanceRequiredPensionFlow-8-12-135%0MoneyMarketInterest0.000.62MoneyMarketBalance0.0012.380.00200320042005Units Purchased(thousands)Cost of BondsBond Cash Flows(per unit)Bond 1Bond 2Bond 3Bond 4-0.98-0.92-0.75-0.801.040.020.030.020.0323.44000BondFlow-22.9724.380.00InitialInvestment30.97>=0>=0>=0all cash figures in $millions0.980.920.750.8 IJ 456789F56789 BondFlow=SUMPRODUCT(B7:E7,UnitsPurchased)=SUMPRODUCT(B8:E8,UnitsPurchased)=SUMPRODUCT(B9:E9,UnitsPurchased)

MoneyMarketBalance=SUM(F7:I7)=MoneyMarketRate*J7=J7+SUM(F8:I8)=MoneyMarketRate*J8=J8+SUM(F9:I9)MoneyMarketInterest

Range NameBondFlowInitialInvestmentMinimumBalanceMinimumRequiredBalanceMoneyMarketBalanceMoneyMarketInterestMoneyMarketRatePensionFlowUnitsPurchasedCellsF7:F9G7L7:L9I2J7:J9I7:I9I1H7:H9B11:E11 3-10 e.Expanded to consider all years through 2012, the spreadsheet is as shown below.PFS should purchase 44.27 thousand units of Bond 1, 51.36 thousand units of Bond 3, and 43.55 thousand units of Bond 4(at a cost of $116.74 million), and invest an additional $8 million in the money market on January 1, 2003.A******819BCDEFGHIJKLMoney Market RateMinimum Required BalanceRequiredPensionFlow-8-12-13-14-16-17-20-21-22-245%0MoneyMarketInterest0.001.771.270.700.001.780.940.001.14MoneyMarketBalance0.0035.3425.4213.990.0035.6718.760.0022.860.******201020112012Units Purchased(thousands)Bond Cash Flows(per unit)Bond 1Bond 2Bond 3Bond 4-0.98-0.92-0.75-0.801.040.020.030.020.031.020.030.031.000.030.030.031.0344.27051.3643.55BondInitialFlowInvestment-116.74124.7447.341.311.311.3152.671.311.3144.860.00>=>=>=>=>=>=>=>=>=>=0000000000all cash figures in $millions IJ F***1516

BondFlow=SUMPRODUCT(B7:E7,UnitsPurchased)=SUMPRODUCT(B8:E8,UnitsPurchased)=SUMPRODUCT(B9:E9,UnitsPurchased)=SUMPRODUCT(B10:E10,UnitsPurchased)=SUMPRODUCT(B11:E11,UnitsPurchased)=SUMPRODUCT(B12:E12,UnitsPurchased)=SUMPRODUCT(B13:E13,UnitsPurchased)=SUMPRODUCT(B14:E14,UnitsPurchased)=SUMPRODUCT(B15:E15,UnitsPurchased)=SUMPRODUCT(B16:E16,UnitsPurchased)

***41516MoneyMarketInterest=MoneyMarketRate*J7=MoneyMarketRate*J8=MoneyMarketRate*J9=MoneyMarketRate*J10=MoneyMarketRate*J11=MoneyMarketRate*J12=MoneyMarketRate*J13=MoneyMarketRate*J14=MoneyMarketRate*J15MoneyMarketBalance=SUM(F7:I7)=J7+SUM(F8:I8)=J8+SUM(F9:I9)=J9+SUM(F10:I10)=J10+SUM(F11:I11)=J11+SUM(F12:I12)=J12+SUM(F13:I13)=J13+SUM(F14:I14)=J14+SUM(F15:I15)=J15+SUM(F16:I16)

Range NameBondFlowInitialInvestmentMinimumBalanceMinimumRequiredBalanceMoneyMarketBalanceMoneyMarketInterestMoneyMarketRatePensionFlowUnitsPurchasedCellsF7:F16G7L7:L16I2J7:J16I7:I16I1H7:H16B18:E18 3-11